UPWARD PLANAR DRAWING OF SINGLE SOURCE ACYCLIC DIGRAPHS. MICHAEL D. HUTTON y AND ANNA LUBIW z

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1 UPWARD PLANAR DRAWING OF SINGLE SOURCE ACYCLIC DIGRAPHS MICHAEL D. HUTTON y AND ANNA LUBIW z Abstract. An pward plane drawing of a directed acyclic graph is a plane drawing of the digraph in which each directed edge is represented as a cre monotone increasing in the ertical direction. Thomassen [24] hasgien a non-algorithmic, graph-theoretic characterization of those directed graphs with a single sorce that admit an pward plane drawing. We present an ecient algorithm to test whether a gien single-sorce acyclic digraph has an pward plane drawing and, if so, to nd a representation of one sch drawing. This reslt is made more signicant in light ofthe recent proof, by Garg and Tamassia, that the problem is NP-complete for general digraphs [12]. The algorithm decomposes the digraph into biconnected and triconnected components, and de- nes conditions for merging the components into an pward plane drawing of the original digraph. To handle the triconnected components weproide a linear algorithm to test whether a gien plane drawing of a single sorce digraph admits an pward plane drawing with the same faces and oter face, which also gies a simpler, algorithmic proof of Thomassen's reslt. The entire testing algorithm (for general single-sorce directed acyclic graphs) operates in O(n 2 )timeando(n) space (n being the nmber of ertices in the inpt digraph) and represents the rst polynomial time soltion to the problem. Key words. algorithms, pward planar, graph drawing, graph embedding, graph decomposition, graph recognition, planar graph, directed graph AMS sbject classications. 68Q20, 68Q25, 68R05, 68R10 1. Introdction. There are a wide range of reslts dealing with drawing, representing, or testing planarity of graphs. Steinitz and Rademacher [22], Fary [10], Stein [21], and Wagner [26] independently showed that eery planar graph can be drawn in the plane sing only straight line segments for the edges. Ttte [25] showed that eery 3-connected planar graph admits a conex straight-line drawing, where the facial cycles other than the nbonded face are all conex polygons. The rst linear time algorithm for testing planarity of a graph was gien by Hopcroft and Tarjan [14]. Planar graph layot has manyinteresting applications, and has been widely stdied as a method to isalize strctres commonly modeled as graphs. Combinational boolean circits, sbrotine call-charts, PERT graphs, isa-hierarchies in AI, and many other objects are natrally described with directed acyclic graphs, and are best nderstood isally when all edges are drawn in the same direction. Planarity is of obios benet in graph-drawing, so it is a natral problem to consider pward drawings in combination with planarity. An pward plane drawing of a digraph is a plane drawing sch thateach directed arc is represented as a cre monotone increasing in the y-direction. In particlar the digraph mst be acyclic (a DAG). A digraph is pward planar if it has an pward plane drawing. Consider the digraphs in Figre 1. By conention, the edges in the diagrams in this paper are directed pward nless specically stated otherwise, and This research was spported in part by NSERC, and performed while the rst athor was at the Uniersity ofwaterloo. A preliminary ersion of the work was presented at the 2nd ACM/SIAM Symposim on Discrete Algorithms (SODA 1991) [16], and an extended abstract later appeared in [27], pp y Department of Compter Science, Uniersity oftoronto, Toronto, Ontario, Canada M5S 1A4. mdhtton@cs.toronto.ca. z Department of Compter Science, UniersityofWaterloo, Waterloo, Ontario, Canada N2L 3G1. albiw@waterloo.ca. 1

2 2 MICHAEL D. HUTTON AND ANNA LUBIW direction arrows are omitted nless necessary. The digraph on the left is pward planar: an pward plane drawing is gien. The digraph on the right is not pward planar thogh it is planar, since placing inside the face f wold eliminate crossings, at the cost of prodcing a downward edge. Kelly [17] and Kelly and Rial [18], and f Upward planar Non-pward-planar Fig. 1. Upward planar and non-pward planar digraphs. also Di Battista and Tamassia [7], hae shown that for eery pward plane drawing there exists a straight-line pward plane drawing with the same faces and oter face, in which eery edge is represented as a straight line segment. This is an analoge of the preiosly mentioned straight-line drawing reslt for ndirected planar graphs. The general problem of recognizing pward planar digraphs has recently been shown to be NP-complete [12]. For the case of single-sorce single-sink digraphs there is a polynomial time recognition algorithm proided by Platt's reslt [19] that sch a digraph is pward planar i the digraph with a sorce-to-sink edge added is planar. An algorithm to nd an pward plane drawing of sch a digraph was gien by Di Battista and Tamassia [7]. For the special case of bipartite digraphs, pward planarity is eqialent to planarity [6]. In this paper we will gie an ecient algorithm to test pward planarity for single-sorce digraphs, eliminating the single-sink restriction. For the most part we will be concerned only with constrcting an pward planar representation enogh combinatorial information to specify an pward plane drawing withot giing actal nmerical coordinates for the ertices. This notion will be made precise in Section 3. We will remark on the extension to a drawing algorithm in Section 7. Or main reslt is an O(n 2 ) algorithm to test whether a gien single-sorce, n-ertex, digraph is pward planar, and if so, to gie a representation for it which leads to a drawing with known methods. This reslt is partly based on a graph-theoretic reslt of Thomassen [24, Theorem 5.1]: Theorem 1.1 (Thomassen). Let ; be aplanedrawing of a single-sorce digraph G. Then there exists an pward plane drawing ; 0 strongly eqialent to (i.e. haing the same faces and oter face as) ; if and only if the sorce of G is on the oter face of;, and for eery cycle in ;, has a ertex which is not the tail of any directed edge inside or on. The necessity of Thomassen's condition is clear: for a digraph G with pward plane drawing ; 0, and for any cycleof; 0, the ertex of with highest y-coordinate cannot be the tail of an edge of, nor the tail of an edge whose head is inside. Since a 3-connected graph has a niqe planar embedding (p to the choice of the oter face) by Whitney's theorem (c.f. [2]) Thomassen concldes that his theorem proides a \good characterization" of 3-connected pward planar digraphs

3 UPWARD PLANAR DRAWING OF SINGLE SOURCE ACYCLIC DIGRAPHS 3 i.e. pts the class of 3-connected pward planar digraphs in NP intersect co-np. An ecient algorithm is not gien howeer (there are potentially an exponential nmber of possible cycles to check), nor does Thomassen address the isse of non-3-connected digraphs (which cold hae an exponential nmber of dierent planar embeddings). The problem ths decomposes into two main isses. The rst is to describe Thomassen's reslt algorithmically we do this in Section 4 with a linear time algorithm, which proides an alternatie proof of his theorem. The second isse is to isolate the triconnected components of the inpt digraph, and determine how topt the \pieces" back together after the embedding of each is complete. This more complex isse is treated in Section 6, after a discssion of decomposition properties in Section 5. The algorithm for splitting the inpt into triconnected components and merging the embeddings of each operates in O(n 2 ) time. Since a triconnected graph is niqely embeddable in the plane p to the choice of the oter face, and the nmber of possible external faces of a planar graph is linear by Eler's formla, the oerall time to test a gien triconnected component is also O(n 2 ), so the entire algorithm is qadratic. 2. Preliminaries. In addition to the denitions below we will se standard terminology and notation of Bondy and Mrty [2]. All digraphs in this paper are acyclic nless otherwise stated, and n always denotes the nmber ofertices in the crrent digraph. We will se the term cycle and the arios notions of connectiity with respect to the nderlying ndirected graph, soa digraph G is connected if there exists an ndirected path between any twoertices in G. For S a set of ertices, GnS denotes G with the ertices in S and all edges incident to ertices in S remoed. If S contains a single ertex we will se the notation Gn rather than Gnfg. G is k-connected if it has at least k +1ertices and the remoal of at least k ertices is reqired to disconnect the graph. By Menger's Theorem [2] G is k-connected if and only if there exist k ertex-disjoint ndirected paths between any two ertices. Asetofertices whose remoal disconnects the graph is a ctset. The terms ct ertex and separation pair apply to ct-sets of size one and two respectiely. A graph which has no ct ertex is biconnected (2-connected). A graph with no separation pair is triconnected (3-connected). For G with ct ertex, a component of G with respect to is formed from a connected component H of Gn by adding to H the ertex and all edges between and H. For G with separation pair f g,acomponent of G with respect to f g is formed from a connected component H of Gnf g by adding to H the ertices and all edges between and ertices of H. The edge ( ), if it exists, forms a component by itself. An algorithm for nding triconnected components 1 in linear time is gien in Hopcroft and Tarjan [13]. A related concept is that of graph/digraph nion: we dene G 1 [ G 2, for components with \shared" ertices to be the inclsie nion of all ertices and edges. That is, for in both G 1 and G 2, the ertex in G 1 [ G 2 is adjacent to edges in each of the sbgraphs G 1 and G 2. Contracting an edge e =( ) inagraphg reslts in a graph, denoted G=e, with the edge e remoed, and ertices and identied. Inserting new ertices within edges of G generates a sbdiision of G. Adirected sbdiision of a digraph G reslts from repeatedly adding a new ertex w to diide an edge ( ) into ( w) and(w ). (Directed) graphs G 1 and G 2 are homeomorphic if both are (directed) sbdiisions of some other (directed) graph. G is planar if and only if eery sbdiision of G is 1 Note that Hopcroft and Tarjan's \components" inclde an extra ( ) edge.

4 4 MICHAEL D. HUTTON AND ANNA LUBIW planar [2]. In a directed graph, the in-degree of a ertex is the nmber of edges directed towards, denoted deg ;. Analogosly the ot-degree (deg + )of is the nmber of edges directed away from. Aertex of in-degree 0 is a sorce in G, andaertex of ot-degree 0 is a sink. Adopting some poset notation: we will write is there is a directed path! of length 0 or more, and <( +!) to emphasize that and are distinct. Vertices and are comparable if or, andincomparable otherwise. If ( ) is an edge of a digraph then dominates, is incident to, and is incident from. 3. A Combinatorial View of Upward Planarity. As discssed by Edmonds and others (see [11]) a connected graph G is planar i it has a planar representation: a cyclic ordering of edges arond each ertex sch that the reslting set of faces F satises 2 = jf j;jej + jv j (Eler's formla). A face is a cyclically ordered seqence of edges and ertices 0 e 0 1 e 1 ::: k;1 e k;1,wherek 3, sch that for any i =0 ::: k; 1 the edges e i;1 (sbscript addition modlo k) and e i are incident with the ertex i and consectie in the cyclic edge ordering for i. We will say that two plane drawings are eqialent if they hae the same representation i.e. the same set of faces. Two plane drawings are strongly eqialent if they hae the same representation and the same oter face. One method of combinatorially specifying an pward planar drawing is proided by the following reslt of (independently) Di Battista and Tamassia [7], and Kelly [17]. They se the concept of a planar s-t digraph, dened to be a planar DAG which has a single sorce s, a single sink t and contains the edge (s t) exactly the pward planarity condition of Platt [19] for single-sorce single-sink digraphs. Theorem 3.1 (Di Battista and Tamassia, Kelly). Let G be a directed acyclic graph. If G is pward planar then edges can be added to it to obtain a planar s-t digraph (i.e. G is a (spanning) sbgraph of a planar s-t digraph). Conersely, if edges can be added tog to obtain a planar s-t digraph G 0,thenG is pward planar. Frthermore, for any planar embedding ; of G 0 with (s t) on the oter face, there is an pward plane drawing of G strongly eqialent to ; with the extra edges remoed. The nal statement was not explicitly gien, howeer to proe their reslt, Di Battista and Tamassia gie an algorithm which takes a planar s-t digraph, nds an arbitrary planar representation of it and otpts an pward plane drawing which respects this embedding, so the statement follows. Their algorithm, which we will reqire later in the paper, rns in O(n) arithmetic 2 steps (O(n log n) arithmetic steps for a straight-line drawing). The disadantage of this NP characterization in terms of planar s-t digraphs is the diclty of testing it. Thomassen's co-np condition on single-sorce digraphs sers from the same problem. For the case of single-sorce digraphs, we will gie a testable (algorithmic) characterization in the next section. To proide some motiation for this algorithm, we gie another characterization of single-sorce pward planar digraphs, eqialent to Thomassen's: First we dene P (), the predecessor set of to be the set f : in Gg. Notice the set P () incldes. DeneG to be indced sbgraph of G on P (). For 2 It is important to specify the time in arithmetic steps, becase the algorithm is necessarily otpt sensitie: coordinates can reqire (n) bits each [8].

5 UPWARD PLANAR DRAWING OF SINGLE SOURCE ACYCLIC DIGRAPHS 5 a planar representation ; of G, dene ; to be the planar representation indced by ;ong. Proposition 3.2. Gien a single sorce DAG G, and a planar representation ; of G with a specied oter face andsorce s on the oter face, G has an pward plane drawing strongly eqialent to ;ithe following condition holds: Condition 3.3. For each ertex 2 V, is a sink on the oter face of the planar embedding ; indced byp (). We will often refer to a planar representation ; satisfying Condition 3.3 as an pward planar representation of G. Since a strongly eqialent pward plane drawing proides the same planar representation ; and predecessors of hae smaller y-coordinates in the drawing, mst be on the oter face of ; 0. Ths the necessity holds. We will complete the proof of this in Section 5 it is not necessary for the algorithm in the next section. 4. Strongly-EqialentUpward Planarity. Consider the following qestion: Gien a single-sorce acyclic digraph G and a planar representation ; for G, withs on the oter face of ;, does G admit an pward planar drawing strongly eqialent to ;? Dene a iolating cycle of G with respect to ; to be a cycle sch thateery ertex of is the tail of an edge inside or on. This is the condition arising from Thomassen's Theorem (1.1). As obsered in the introdction, a iolating cycle in ; precldes the existence of an strongly eqialent pward drawing. We present a linear time algorithm to test whether G has an pward planar embedding strongly eqialent to ; with a designated oter face. The algorithm will retrn the edges necessary to agment G so that sinks occr only on the oter face in the positie case, or a iolating cycle in the negatie case. Since any planar representation of a single-sorce DAG with the sorce and all sinks on the oter face is a sbgraph of a planar s-t digraph simply designate one sink as t, and add an edge from the sorce and all other sinks to it the algorithm proides a new proof of Thomassen's theorem. The algorithm is recrsie, and the proof that it works is by indction. If there is a sink on the oter face of ;, then recrsiely (triial if G has one node) determine a iolating cycle for Gn (in which casewe are done) or a set of edges X reqired to agment;n (Gn) to a planar representation ; 0 with all sinks on the oter face. Now add and edges incident to to the oter face of ; 0. To determine the additional reqired edges to resole the internal sinks in the new faces, consider all ertices w which are sinks on the oter face of ; 0, bt are not sinks on the oter face of ;. Adding the edges (w ) (where they do not already exist in G) tox retains planarity, single-sorcedness and acyclicity ing [ X and does not change the oter face. It remains to deal with the case when the oter face of ; has no sink. We claim that in this case G has a iolating cycle: If the oter face of ; is a cycle then it is a iolating cycle. If the oter face is a walk, then follow itstartingats, andlet be the rst ertex which repeats. Vertex mst be a ct ertex. Consider the segment of the walk from to. If this segment contains only one other ertex, say, then is a sink, contradiction. Otherwise we obtain a cycle C from to. The two edges incident with mst be directed away from and no other ertex is a sink on C, so C mst be a iolating cycle. The aboe algorithm can be implemented in linear time (so that each ertex is inoled in no more than a constant nmber of operations), sing data strctres no more complicated than a linked list. We then hae

6 6 MICHAEL D. HUTTON AND ANNA LUBIW Theorem 4.1. Gien an n-ertex single-sorce acyclic digraph G and a plane representation ;, the aboe algorithm tests, in linear time, whether G admits an pward planar drawing strongly eqialent to ;. 5. Decomposition properties of Upward Planar Graphs. This section completes the discssion of pward planar representations and introdces arios decomposition-properties of pward planar digraphs. The prpose is twofold: rstly, the properties are necessary for the proofs in the next section secondly, they proide an intitie look at the strctre of pward planar digraphs, and hence motiate the decomposition approach we take in the recognition algorithm. We begin by completing the proof of Proposition 3.2 from Section 3: Proof. (Sciency) We need that for any planar representation ; of G satisfying Condition 3.3, G admits an pward plane drawing strongly eqialent to ; eqialently, that the existence of a iolating cycle precldes Condition 3.3 from holding for some ertex. Sppose a iolating cycle exists in G with respect to ;. Let G be the sbgraph of G formed by edges and ertices inside or on. Withot loss of generality, G has one sorce s, which mst lie on : If G had two sorces s 1 and s 2, then both wold be on. Since G has a single sorce s, there exist directed paths P 1 form s to s 1 and P 2 from s to s 2. The last edge of each path is not in G. If either path has a ertex other than its terminal ertex on, then adding to the portion of the path from the last sch ertex to the terminal prodces another iolating cycle enclosing a larger sbgraph with one fewer sorces. Otherwise, P 1 and P 2 contain a last common ertex, and adding the portions of the paths from that ertex to the terminals s 1 and s 2 prodces another iolating cycle enclosing a larger sbgraph with one fewer sorces. Ths we can assme that G has a single sorce s. The remainder of the proof references Figre 2. P2 y x z G P 1 Σ... P 3 s Σ Fig. 2. Violating cycle precldes Condition 3.1. Starting from s,walk conter-clockwise arond. Let x be the rst encontered ertex with both edges of directed towards x. Lety be the rst encontered ertex after x with both edges of directed away from y. Note that both exist, thogh y may bes. Let P 1 be the directed path from s to x, conter-clockwise on, and let P 2 be the directed path from y to x clockwise on. Since s is the single sorce of G, there is a directed path P 3 in G from s to y. (If y = s then P 3 is this single ertex.) P 3 cannot contain a ertex of P 2 other than y otherwise we wold get a directed cycle sing portions of P 2 and P 3. Let be the last ertex of P 1 on P 3. Let be the simple ndirected cycle consisting of the portion of P 3 from to y, the portion of P 2 from y to x, and the portion of P 1 from

7 UPWARD PLANAR DRAWING OF SINGLE SOURCE ACYCLIC DIGRAPHS 7 to x. LetG be the sbgraph of G formed by edges and ertices inside or on. Since is a iolating cycle, x is not a sink in G, so there is an edge (x z) inside, and ths inside. We will show that ertex z iolates condition 3.1. Vertex z cannot be on otherwise a directed cycle is formed. Ths z is strictly inside. Bt all the ertices of are predecessors of z. Ths z iolates condition 3.1. Note that the reslts of the preceding two sections, combined with the characterization of Di Battista and Tamassia, and the single-sorce characterization of Thomassen gie: Theorem 5.1. The following conditions are eqialent for a single-sorce DAG G with planar representation ; haing a designated oter face and single sorce s which is on the oter face: (i) G has an pward plane drawing strongly eqialent to ;. (ii) G is a (spanning) sbgraph of some planar s-t digraph which has an pward plane drawing strongly eqialent to ; (after remoal of the extra edges) (iii) for all 2 G, is a sink on the oter face of;. (i) ; does not contain a iolating cycle. We note that condition (iii) is the only one which canobiosly be tested in polynomial time. In the remainder of this section we gie some operations which presere pward planarity. The rst operation contracts an edge connected to a ertex of in- (ot-) degree 1. The second attaches one pward planar digraph to another at a single ertex. The third attaches an pward planar digraph in place of an edge of another pward planar digraph. The last splits a ertex into two ertices. First we will proe a sefl preliminary reslt: Proposition 5.2. Let G be a connected pward planar digraph. Then G is a sbgraph of some single-sorce pward planar G sch that all non-sorce 2 V (G) hae the same in-degree ing as in G. Proof. We illstrate how to add the edges reqired to \resole the extra sorces" withot aecting the in-degree of non-sorce ertices. Let ; be a drawing of G in the plane bonded by x min, x max, y min and y max, with height h, width w and centred at (0 0). Withot loss of generality, we assme ; is a straight linedrawing. Add new ertices s, t, l and r at (0 ;2h) (0 2h) (;2w 0) and (2w 0) respectiely. Add lines (edges) (s l) (s r) (r t)and(l t). Add frther edges (s w) for all ertices w drawn with y-coordinate of y min and (w t) for all ertices w drawn with y-coordinate of y max. The constrction so far has merely added a specied oter face on the drawing, with a niqe maximm sink and minimm sorce, so clearly the reslting drawing ; (digraph G )isanpward plane drawing (pward planar digraph). We now wish, for each sorce x, to \resole" the sorce by adding a new edge incident toit the reslting digraph will proe or proposition. Let ; be an pward plane drawing of G, and perform the following operation for each sorce x, except the one jst added in the oter face: extend a line L ertically downwards from x to the rst line or ertex in the drawing. If L rst intersects a ertex w, add the (w x) edge to both G and the drawing ; the reslt is clearly pward planar. If L rst intersects an edge, rotate it along the edge ntil L hits some ertex w of ; and add the edge (w x) as before for one of the two directions, a ertex will be fond before the line becomes horizontal. Neither operation added in-degree to a non sorce ertex, so the claim is satised. Note that the aboe is of no particlar algorithmic signicance, since the drawing

8 8 MICHAEL D. HUTTON AND ANNA LUBIW of G (or existence thereof) is the goal rather than the inpt. Howeer, it allows s to proe the following lemmas in the more general context of pward planar digraphs, i.e. withot the single-sorce assmption. Lemma 5.3. Let G be a DAG and, dominated by, be a ertex of G with in-degree 1. Then, G=( ) is pward planar if G is pward planar. (See Figre 3(a).) Note that the same reslt holds for G and edge ( ) with deg + =1bysymmetry. Lemma 5.3 is a generalization of a preiosly known fact that G is pward planar i any directed sbdiision of G is (c.f. [24]). Proof. Let; beanpward plane drawing of G. Applying proposition 5.2, there is a single-sorce digraph G containing G as a sbgraph in which the in-degree of is still 1. Clearly if the reslt holds for G =( ) it holds for any sbgraph, namely G=( ), so we it will be scient to assme G is a single-sorce digraph for the remainder of the proof, and show Condition 3.3 holds. Let ; be a planar representation for G, with a designated oter face, satisfying Condition 3.3. Let ; w for w 2 V be as dened for Condition 3.3. Then ; 0, formed by contracting ( ) in ;, is a planar representation for G 0 = G=( ) with a designated oter face. Clearly if some w 6= is on the oter face of ; w,itisontheotherface of ; w =( ). This, with the fact that G is acyclic, implying G w =( ) =G 0 w for all w 2 V ;fg (i.e. P(w) doesn't change as a reslt of contracting ( )), gies that w is a sink on the oter face of ; 0 w Condition 3.3. Topologically this constrction can be iewed as \plling" and all edges incident from down a corridor of width arond ( ) ntil and meet. f (a) (b) e e e e3 e e (c) " e e e e (d) e e 4 Fig. 3. Properties of Upward Planar Representation. Lemma 5.4. Let G be an pward planar digraph with a ertex, andleth be an pward planar digraph with a single sorce 0. Let G 0 be the digraph formed by identifying and 0 in G [ H. Then G 0 is pward planar. (See Figre 3(b).) Proof. Asaboe, there is an pward planar single-sorce digraph G containing G by proposition 5.2, and G 0 is a sbgraph of G [ H with and 0 identied, so it is scient to proe the reslt for a single-sorce pward planar G. Sppose ; G and ; H are the gien planar representations, with designated oter faces, both satisfying Condition 3.3. Let ; G and ; H be as dened for Condition 3.3 for G and H respectiely. Weshowhow to constrct a planar representation ; 0 for

9 UPWARD PLANAR DRAWING OF SINGLE SOURCE ACYCLIC DIGRAPHS 9 G [ H (identifying and 0 ), with a gien oter face, which satises Condition 3.3. If is on the oter face of ; G, then place ; H in the oter face, identifying and 0. Otherwise, there are (possibly) k + 1 faces of ; G corresponding to the oter face of ; G (for k the ot-degree of ) insert ; H in any one of these faces. It is easy to show that nder this constrction w is a sink on the oter face of ; 0 w if it was a sink on the oter face of ; G w (respectiely ; H w ) preiosly. This constrction can be iewed as \inserting" the drawing for H into some face \aboe" in the drawing of G. Lemma 5.5. Let G be anpward planar digraph with an edge ( ), andh be an pward planar digraph with a single sorce 0 and a sink 0 both on the oter face. Let G 0 be the digraph formed byremoing the ( ) edge of G and adding H, identifying ertex with 0 and ertex with 0. Then G 0 is pward planar. Proof. This has the same aor as the preios proof, so we can be more brief. Again, by Proposition 5.2 it is scient to assme that G has a single sorce. Let ; G and ; H be planar representations, with designated oter faces, satisfying Condition 3.3. Form a planar representation ; 0 of G 0,by replacing ( ) by; H in ; G. The reslt is planar, and has a well-dened oter face. We need that ; 0 satises Condition 3.3. As in the preios proof, it is easy to show thatw is a sink on the oter face of ; 0 w wheneer it is a sink on the oter face of ;G w (respectiely ;H w ). This constrction can be iewed as replacing a directed edge in an pward plane drawing of G with another pward plane drawing of H which is, in some sense, \topologically eqialent" to an edge within the drawing of G. Lemma 5.6. Let G be a DAG which has an pward planar representation where the cyclic edge order abot ertex is e 0 ::: e k;1 (ertices 0 ::: k;1 ). Let G 0 be the DAG formed by splitting into two ertices: 0 incident with edges e i ::: e j,and 00 incident with edges e j+1 ::: e i;1 (i 6= j, arithmetic mod k). Then G 0 is pward planar. If G hadasinglesorce, and i and j are sch that each of 0 and 00 retain at least one incoming edge, then the reslting G 0 is also a single-sorce digraph. (See Figre 3(d).) Proof. The last statement is clearly tre no new sorces can be added by the constrction if each newertex has an incoming edge. Again it is scient toshow the rst part for single-sorce G, since the reslting digraph is otherwise a sbgraph of the constrction applied to G (of Proposition 5.2). Let ; be a planar representation for G satisfying Condition 3.3. Withot loss of generality, assme that the constrction does not make 0 a sorce nless was, itself, a sorce. We proe the reslt for G 00 = G 0 +( 0 00 ) it is easy to agment ; 0 (the planar representation formed by separating into 0 and 00 in ;) to ; 00 with the edge ( 0 00 )as 0 and 00 share a face. The constrction of ; 00 from ; preseres planarity and cannot introdce a dicycle it remains to show Condition 3.3 holds for G 00 and ; 00. The set of faces in ; 00 is is identical to that of ;, sae forthe two new faces sharing ( 0 00 ), so Condition 3.3 is satised for all w not incident from either 0 or 00. Any i incident from 0 or 00 (ia edge e i ) is clearly on the oter face of ; 00 i wheneer it is on the oter face of ; i, since the constrction can only add ertices to an oter face, neer remoe them. 6. Separation into Tri-Connected Components. The algorithm of Section 4 tests for pward planarity of a single-sorce DAG G starting from a gien planar representation and oter face of G. In principle, we cold apply this test to all planar representations of G, bt this wold take exponential time. In order to aoid this, we

10 10 MICHAEL D. HUTTON AND ANNA LUBIW will decompose the digraph into biconnected and then into triconnected components. Each triconnected component has a niqe planar representation (see [2]), and only a linear nmber of possible oter faces. We can ths test pward planarity of the triconnected components in qadratic time sing the algorithm of Section 4. Since we will perform the splitting and merging of triconnected components in qadratic time, the total time will then be qadratic. To decompose G into biconnected components we se: Lemma 6.1. ADAG G with a single sorce s and a ct ertex is pward planar i each of the k components H i of G (with respect to ) is pward planar. Proof. IfG is pward planar then so are its sbgraphs the H i 's. For the conerse, note that if 6= s then is the niqe sorce in all bt one of the H i 's and if = s then is the niqe sorce in each H i. Apply Lemma 5.4. Diiding G into triconnected components is more complicated, becase the ct-set ertices impose restrictie strctre on the merged digraph. In the biconnected case, it is scient to simply test each component separately, since biconnected components do not interact in the combined drawing. The analogos approach for triconnected components wold be to add a new edge between the ertices of the ctset in each component, then perform the test recrsiely. This, howeer, does not sce for pward planarity, as illstrated by the two examples in Figre 4. (Recall or conention that direction arrow-heads are assmed to be \pward" nless otherwise specied.) In (a), the nion of the digraphs is pward planar, bt adding the edge ( ) to each makes the second component non-pward-planar. In (b), the digraph is non-pward-planar, bt each of the components is pward planar with ( ) added. (a) (b) Fig. 4. Added complication of 2-ertex ct-sets. We will nd it conenient to split the digraph G into exactly two pieces at a separation pair f g, where one of these pieces, E, is a component with respect to the separation pair, and the other piece, F, is the nion of the remaining components. This forces each piece to t into one face of the embedding of the other piece: Lemma 6.2. For G, E F as aboe, let ; be a plane embedding of G, andlet; E and ; F be the embeddings indced one and F,respectiely. Then in ;, allofe lies in a single face of; F, and all of F lies in a single face of; E. Frthermore, at least one of E, F, mst lie in the oter face of; F ; E,respectiely. Proof. Any distinct ertices x and y in E, neither being or, mst share a path in E which aoids both and (lies entirely within E). Hence, for ; a plane embedding of G and ; E and ; F the respectie sb-embeddings of E and F,ifertices x and y of E are in dierent faces of ; F they cold not share a path which aoids both and withot iolating planarity. Clearly, also, one of E, F mst hae two ertices on the oter face of the total drawing ; (which has at least 3 ertices) and hence mst lie entirely in the oter face of the other sb-drawing. We will test pward planarity of a biconnected digraph G by breaking it at a ct-set into pieces E and F as aboe, and looking for pward planar embeddings ; E

11 UPWARD PLANAR DRAWING OF SINGLE SOURCE ACYCLIC DIGRAPHS 11 and ; F that t together as in Lemma 6.2. We needafacein; E that contains and and is the right \shape" to accommodate the \shape" of ; F and we need a face in ; F that contains and and is the right \shape" to accommodate the \shape" of ; E. (Figre 4(b) showed an example where these conditions fail.) These conditions will be enforced by adding a \marker" connecting and to E (F, respectiely) that captres the \shape" of ; F (; E,respectiely), and forces and to lie in a common face. For example, the simplest case is when is the sorce and is the sink of F then the marker representing ; F in E is a single ( ) edge. Besides playing the primary role described aboe, the markers will also be sed to make the two components 3-connected, and single-sorce, ths allowing s to recrse on smaller sbproblems. The markers we are interested in are shown in Figre 5. w t w t w s M M M s t M t Fig. 5. Marker Graphs. We need one other main idea. The last statement of Lemma 6.2 is that one of E, F,mst lie in the oter face of ; F,; E, respectiely. For ndirected digraphs this cases no problem, since any face can be made the oter one. Howeer, for pward planarity, this condition complicates things. The sitation is simplied when s 6=. In this case we willtake E to be the f g component containing s, and so ; E mst lie in the oter face of ; F. When s 2f g we mst do extra work to decide the \oter" component. Haing determined or decided that ; E mst lie in the oter face of ; F,we know that and mst be on the oter face of ; F.Ths or algorithm will sole the more general problem of testing pward planarity nder the condition that some specied set X of ertices, called the \oter" set, mst lie on the oter face. To smmarize, gien a biconnected digraph G and an \oter" set of ertices X, we break G at a ct set f g into one component E containing s, and the nion of the remaining components F.We add appropriate markers to E and F, specify their \oter" sets, and recrse. We mst proe that G has an pward planar embedding with its \oter" set on the oter face i the smaller digraphs do. The details and proofs of this plan make p the remainder of this section. We will consider three cases separately: when and are incomparable when and are comparable with s<< and when and are comparable with = s. An important note to make at this time is that the markers, except for M,are sbgraphs attachedatonlytwo ertices, which means that f g will still constitte a ct-set. For the prposes of determining ct-sets, and making recrsie calls, the markers shold be treated as distingished edges a single edge labelled to indicate its role. As long as the type of marker is identied, the algorithm can contine to treat the ertices of attachment as sorce, sink or neither, as appropriate for the particlar operation Ct-set f g and are incomparable. Here we consider ertex ctsets f g which are incomparable (then neither is s). We diide the digraph G

12 12 MICHAEL D. HUTTON AND ANNA LUBIW at f g into two sbgraphs the sorce component E (the one component which contains the sorce s), and the nion of the remaining components F. First we need some preliminary reslts: Proposition 6.3. If G is a connected DAG with exactly two sorces and, then there exists some w t sch that two ertex disjoint (except at w t )directed paths +!w t and +!w t exist in G. Proof. Let G be sch adag and let P be an ndirected path from to. Note that eery x in P is comparable with either or, otherwise G has more than two sorces. Follow P from to the rst node x (following y on P ) incomparable with (in G). Then x is comparable with and (x y) is an edge in G (otherwise <x), so y is also comparable with. Taking the rst common ertex in the paths +!y and +!y gies w t. The following reslt shows the existence of lower bonds and pper bonds (in the partial order corresponding to G) nder certain conditions. This allows s to proe the necessity conditions in Theorem 6.5 (to come). Lemma 6.4. If G is a biconnected DAG with a single sorce s,and and are incomparable ertices in G, thenthere exists some w s sch that two ertex disjoint (except at w s ) directed paths w s +! and w s +! exist in G. If f g is a ct-set in G, then there also exists some w t sch that two ertex disjoint (except at w t )directed paths +!w t and +!w t exist in G. Proof. SinceG is a single sorce digraph, there exist directed paths from s to and s to in G. Taking the last common ertex in these paths gies w s. For the existence of w t,let and be an incomparable separation pair of G. Since f g cts G into at least two connected components, any non-sorce component H has and as its (exactly) two sorces, and the reslt follows from Proposition 6.3. We are now ready to proceed with the statement of the rst main reslt of the decomposition. Theorem 6.5. Let G be a biconnected directed acyclic digraph with a single sorce s and let X = fx i gv (G) be a set of ertices. Let f g be a separation pair of G, with and incomparable. Let E be theconnected component of G with respect to f g containing s, andf be the nion of all other components. Then, G admits an pward plane drawing with all ertices of X on the oter face if and only if (i) E 0 = E [ M t admits an pward plane drawing with all ertices of X in E on the oter face, and w t on the oter face if some x 2 X is contained inf. (ii) F 0 = F [ M s admits an pward plane drawing with all ertices of X in F on the oter face. Here, as in the remaining cases, the proof will hae the same basic aor. The necessity of the marker-conditions will follow from the existence of the corresponding marker `within' (i.e. homeomorphic to a sbgraph of) the companion component. The sciency will be shown by applying the properties of an pward planar representation from Section 4 to combine pward planar representations for the two sbproblems into a single pward planar representation. Proof. (Necessity): Sppose G admits an pward planar drawing (representation) ; with all x i 2 X on the oter face. Follow Figre 6. Since and are incomparable, there exists a w s and ertex-disjoint directed paths w s +! and w s +! in G by Lemma 6.4 specically, these mst be in E if G has a single sorce. Then F 0 = F [f(w s ) (w s )g is homeomorphic to a sbgraph

13 UPWARD PLANAR DRAWING OF SINGLE SOURCE ACYCLIC DIGRAPHS 13 of G and hence pward planar itself. F 0 can be obtained from G by deleting and contracting E to its marker this will not decrease the set of ertices on the oter face. Ths, since ; has all the x i 's of F on its oter face, F 0 has an pward planar drawing with all ertices of X in F on the oter face. We hae (ii). Similarly, there exists some w t in F sch that E [f( w t ) ( w t )g is homeomorphic to a sbgraph of G, soe 0 is pward planar. As aboe, any x i in E and also on the oter face of ; will be on the oter face of the sbdrawing formed by ; E and the +! w t +!w t paths. By Lemma 6.2, ; F mst lie entirely within one face of ; E,so if some x i in F is on the oter face of the drawing ;, then the portion of the drawing formed by the +!w t and +!w t paths (hence w t itself) mstbeintheoterfaceof the sb-drawing ; E, giing (i). E F ws x 1 w t x 2 x k x 1 F w s x 2 E x3 w t x k (a) No x i's in F (b) x i's in F Fig. 6. Merging E and F ct-set f g incomparable. (Sciency): Sppose E 0 and F 0 admit pward planar representations satisfying (i) and (ii). Identifying the single sorce w s of F 0 and w t in E 0 (call the new ertex w) as per Lemma 5.4, the reslt G 0 is pward planar. Splitting w into w l with the leftmost two ertices and w r with the rightmost two ertices (Lemma 5.6), and contracting the (w l ) and (w l ) edges (Lemma 5.3) gies exactly G, whichishencepward planar. The constrction is illstrated in Figre 7. ws wt Lemma 5.2 w Lemma 5.4 Lemma 5.1 Fig. 7. Merge constrction f g incomparable. For the sciency of the x i conditions, we notice that all non-marker ertices of X on the oter face of the E 0 drawing are also on the oter face of the constrcted drawing. If F contains no x i this is scient otherwise w t being on the oter face of E 0 garantees that the non-marker ertices of X in F 0 are also on the oter face of the reslt Ct-set f g, where <, 6= s. Here we consider any other ertex ct-sets not inoling the sorce s. We again diide G into the sorce component E and the nion of the remaining components F. Note that can be a sorce in E, as long as there is a to path in F.

14 14 MICHAEL D. HUTTON AND ANNA LUBIW An additional preliminary reslt will be sefl. Lemma 6.6. If G is a biconnected DAG with a single sorce s and ct-set f g, where < in G and 6= s, then in any non-sorce component H of G with respect to f g, where deg + >0, there exists some w t sch that +! w t and +! w t are ertex disjoint directed paths in H. Proof. Noertex other than and can be a sorce in H, otherwise G has more than one sorce and is always a sorce in H. If is also a sorce, then we are done by Proposition 6.3. If is not a sorce, let w 2 H be a ertex dominated by. G is biconnected, so there are two ertex disjoint! + w ndirected paths in G. Bt and are ctertices in G, so at least one of the paths P lies completely within H and does not contain (as w is in H and the only exit points from H are and ). Eery x on P is comparable with either or, orelseg has more than one sorce. Find the last ertex y on P which has a! + y path (in G) withot. If y = w, thenweare done. Otherwise, the ertex x following y on P has any!x + path necessarily going throgh. Then there exist directed paths!x, +!x + with the latter not containing so the rst common ertex on these paths proides a w t. We cannowcontine with the second main reslt of the decomposition. Theorem 6.7. Let G be a biconnected directed acyclic digraph with a single sorce s, and let X = fx i gv (G) be a set of ertices. Let f g be a separation pair of G with <in G and 6= s. Let E be the sorce component of G with respect to f g and F be the nion of all other components. Then, G admits an pward plane drawing with all ertices of X on the oter face if and only if (i) E 0 =(E [ F -marker) admits an pward plane drawing with all ertices of X in E on the oter face andw t (if it exists, otherwise the edge ( )) on the oter face if some x 2 X is contained inf. (ii) F 0 =(F [ E-marker) admits an pward plane drawing with w t (if it exists, otherwise the edge ( )) and all ertices of X in F on the oter face. where and F -marker = 8 < : M t M M t if is a sorce inf if is a sink in F otherwise. Mt E-marker = M if is a sorce ine otherwise. Proof. (Necessity) Sppose G admits an pward plane drawing with all x i 2 X on the oter face. Follow Figre 8. (Necessity of condition (i)): If is a sorce in F, then there exists some w t in F and ertex disjoint paths +! w t and +! w t by Proposition 6.3 so E 0 = E [ M t is homeomorphic to a sbgraph of G and is pward planar. If is a sink in F,then is the single sorce of F,asonly and are possible sorces. Ths, in F, there is a path +!, soe 0 = E [ M is homeomorphic to a sbgraph of G and is pward planar. If is neither a sorce nor a sink in F then, by Lemma 6.6, there is also some w t >and disjoint directed paths +! w t and +! w t in G. Since is a non-sorce

15 UPWARD PLANAR DRAWING OF SINGLE SOURCE ACYCLIC DIGRAPHS 15 in F, there is also a +! path in F. This path crosses the +! w t path at some latest ertex z on that path, so E [ (! z) [ (z +! ) [ (z +! w t ) [ ( +! w t )isa sbgraph of G and hence pward planar. Note that these for paths are disjoint. Since z has in-degree one we can contract the! z path to withot destroying pward planarity,by Lemma 5.3, so E [f( ) ( w t ) ( w t )g has an pward planar sbdiision and is pward planar itself. By Lemma 6.2 F lies in a single face of ; E, so no other ertices lie inside the w t triangle, and the extra edges and ertex for M t can be added withot destroying planarity 3. If some x i is in F then, by the same argment as Theorem 6.5, all of F mst be in the oter face of ; E. The marker, hence w t or the ( ) edge as appropriate, is therefore in the oter face of the drawing indced by E on ;. (Necessity of condition (ii)): If is a sorce in E, then, by Proposition 6.3, there are ertex disjoint paths s +! w t and +! w t in E. There mst be an s +! path in E, otherwise there is either a second sorce ( is a sorce in F, so it cannot also be a sorce in E) or a cycle in G ( < in G, so there can be no +! directed path in E). Let z be the last ertex common to paths s +! and s +! w t. Then, F [f(z ) (z w t ) ( w t )g is homeomorphic to a sbgraph of G and is pward planar. Since deg ; = 1 (in this digraph), the edge (z ) can be contracted withot destroying pward planarity, by Lemma 5.3, and F 0 = F [ M t is pward planar. Otherwise ( a non-sorce), if <in E, thenf 0 = F [ M is homeomorphic to a sbgraph of G and, hence, is pward planar. If and are incomparable in E, then they share a greatest lower bond w s,by Lemma 6.4, and F [f(w s ) (w s )g is pward planar. Again, deg ; =1inF, so the (w s ) edge can be contracted to gie F 0 = F [ M. The reqirement for the E-marker to be on the oter face of ; indced by F follows as before: ; E lies entirely within one face of ; F, and this is necessarily the oter face since E contains the sorce s. (Sciency) Sppose E 0 and F 0 admit pward plane drawings meeting the reqirements (i) and (ii). Case 1: is a sorce in F : (See Figre 8(a).) If is a sorce in F it cannot at the same time be a sorce in E, as<in either E or F.Ths F 0 = F [( ) ispward planar with single sorce. Using Lemma 5.4, add F 0 (with and renamed as 0 and 0 )toe 0, identifying 0 with w t.we can do this so that edges ( w t ) and (w t 0 ) are consectie in the cyclic order abot w t. Using Lemma 5.6, split w t by making these two edges incident withanewertex 1 and the remaining edges incident with anewertex 2.Now 0 and 1 hae in-degree 1, so se Lemma 5.3 to contract their in-edges, ths identifying and 0.Vertex 2 has in-degree 1 so contract ( 2 ). The reslt is the digraph G, and ths G is pward planar. Case 2: is a sink in F : (The two possibilities are illstrated in Figre 8 (b) and (c).) If is a non-sorce in E, thenf 0 = F [ ( ) ispward planar with and on the oter face by assmption. If is a sorce in E, thenf 0 = F [ M t is pward planar with w t on the oter face. In either case F is pward planar with single sorce and sink on the oter face. By Lemma 5.5 we can add F to E 0 in place of the ( ) edge in E 0, and the reslt, G, ispward planar. Case 3: is a non-sorce/sink in F : (See Figre 8(d).) Sppose is a sorce in E. Then F 0 = F [ M t is pward planar with the sink w t on the oter face. Using Lemma 5.5, add F 0 (renaming and to 0 and 0 respectiely) to E 0 in place of the 3 The point of adding these edges is to x the face in E for the sciency conditions.

16 16 MICHAEL D. HUTTON AND ANNA LUBIW F E w t F E (a) a sorce in F (b) asinkinf, non-sorce in E F wt E F w t w t E (c) asinkinf, sorce in E (d) a non-sorce/sink in F, sorce in E F E wt (e) a non-sorce/sink in F, non-sorce in E Fig. 8. Merging E and F ct-set f g and <. edge ( ), identifying 0 with and w t with. Throw away the edge ( w t ) and the remaining marker edges of E 0.Vertex now has in-degree 1 so the edge ( 0 )canbe contracted by Lemma 5.3, and the reslt, G, ispward planar. Note that the M t marker attached to E 0 is stronger than we actally reqire here (M wold do), bt it necessarily does exist (as preiosly proen) and is needed for the next part of this case. Sppose then that is a non-sorce in E. Consider the pward planar representation of E 0 and throw away the marker edges, sae for( w t ) ( w t ) ( ), which then form a face. F 0 = F [( ) ispward planar with and on the oter face. Let z be some sink on the oter face, and add the edge ( z) toobtainf 00,pward planar with z on the oter face. Using Lemma 5.5, add F 00 (with and renamed to 0 and 0 )toe 0 in place of the edge ( w t ), identifying 0 with and z with w t.do this so that 0 and share the face of edges ( 0 ) ( 0 z) ( ) ( z). Clearly we can

17 UPWARD PLANAR DRAWING OF SINGLE SOURCE ACYCLIC DIGRAPHS 17 now identify the ertices and 0. We obtain an pward planar digraph containing G as a sbgraph. See Figre 8(e). As in the proof of Theorem 6.5 we notice that all ertices on the oter face of E 0 are necessarily on the oter face of the combined drawing, and if some x i exists in F,thenitisontheoterfaceofF 0, and is forced to the oter face of the combined drawing by the second part of condition (i) Ct-set fs g. As mentioned in the introdction to Section 6 (see also Lemma 6.2), it is important to be able to distingish the \inner" and \oter" components. The inner component will be embedded in a face of the oter one, and ths the inner component willhae tohae the marker on its oter face since this marker is aproxy for the oter component. If we hae tocheck each component as a potential inner component, we mst recrsiely soletwo sbproblems for each component, and an exponential time blowp reslts. Until now, the oter component has been niqely identied as the sorce component, since that component cannot lie within an internal face of any other component. If we hae a ct-set of the form fs g where s is the sorce, then we lose this restriction, so we handle it instead by reqiring one of the components, E, to be 3-connected so that deciding if it can be the inner face does not reqire recrsie calls. To decide if E can be the inner face we need to test if it satises the role of E in the preios theorem i.e. has an pward planar representation with the marker on its oter face. This can be done in linear time sing the algorithm of Section 4. If G has only ct-sets of the form fs g, then, for at least one sch ct-set, one of the components will be triconnected. Gien the list of ct-sets we can nd sch a ct-set and sch a component in linear time sing depth-rst search. We captre these ideas in terms of two theorems. One is applicable if the triconnected component, E, can be the inner component, and one if it cannot. E \can be" the inner component if and only if it satises the same conditions that the inner component F satised in the preios Theorem 6.7. The similarity of both of these theorems to Theorem 6.7 shold be clear. Note that in the statement of these theorems, we contine to se (redndant since = s) for consistency with preios sage. Theorem 6.8. Let G be a biconnected DAG with a single sorce s, and let X = fx i gv (G) be a set of ertices. Let f g be a separation pair of G where = s, E be a 3-connected component of G with respect to f g, andf be the nion of all other components of G with respect to f g. If E 0 =(E [ F -marker) admits an pward plane drawing with w t (if it exists, otherwise the edge ( )) and all ertices of X in E on the oter face, then G admits an pward plane drawing with all ertices of X on the oter face if and only if (i) where and F 0 =(F [ E-marker) admits an pward plane drawing with all ertices of X in F on the oter face, and w t (if it exists, otherwise the edge ( )) also on the oter face if some x 2 X contained ine. E-marker = 8 < : M t M M t if is a sorce ine if is a sink in E otherwise.