On terminal delta wye reducibility of planar graphs

Transcription

1 On terminal delta wye reducibility of planar graphs Isidoro Gitler 1 Feliú Sagols 2 Departamento de Matemáticas Centro de Investigación y de Estudios Avanzados del IPN Apartado Postal México City, D.F. igitler@math.cinvestav.mx fsagols@math.cinvestav.mx Abstract A graph is terminal Y -reducible if it can be reduced to a set of terminal vertices by a sequence of series-parallel reductions and Y -transformations, where terminals are distinguished vertices which cannot be deleted by reductions and transformations. Reducibility of terminal graphs is very difficult and in general not possible for graphs with more than three terminals (even planar graphs). Terminal reducibility plays an important role in decomposition theorems in graph theory and in important applications, as for example, network reliability. We prove terminal reducibility of planar graphs with at most three terminals. The most important consequence of our proof is that it implicitly gives an efficient algorithm, of order O(n 4 ), for reducibility of planar graphs with at most three terminals that also can be used for restricted reducibility problems with more terminals. It is well known that these operations can be translated to operations on the medial graph. Our proof makes use of this translation in a novel way, furthermore terminal vertices are now seen as terminal faces and by duality of the reductions and transformations, the set of terminals can be taken as a set of vertices and a set of faces of the original graph. Key words and phrases. Delta-Wye transformations, medial graph, planar graphs and topological graph theory. 1 Background A graph is planar if it can be drawn in the plane without edges crossing, and it is a plane graph if it is so drawn in the plane. The drawing separates the rest of the plane into regions called faces. We consider graphs which may include loops, edges whose two end vertices are identical, and parallel edges, two edges with the same end vertices; we also consider parallel loops. Y Operations: A class of graphs Q, is said to be ( Y) reducible to a canonical simple graph structure P, if any G Q can be reduced to P, by repeated application of the following four reductions and two transformations : R0 Loop reduction. Delete a loop. R1 Degree-one reduction. Delete a degree one vertex and its incident edge. R2 Series reduction. Delete a degree two vertex y and its two incident edges xy and yz, and add a new edge xz. R3 Parallel reduction. Delete one of a pair of parallel edges. Each of these reductions decreases the number of vertices or edges in a graph. Two other transformations of graphs are important. A wye (Y) is a vertex of degree three. A delta ( ) is a cycle {x, y, z} of length three. The transformations are: 1 Corresponding author. Partially supported by CONACyT grants U40201, and SNI. 2 Partially supported by CONACyT grants U40201, and SNI. 1

2 Y Wye-delta transformation. Delete a wye w and its three incident edges wx, wy, wz, and add in a delta {x, y, z} with edges xy, yz and zx. Y Delta-wye transformation. Delete the edges of a delta {x, y, z}, add in a new vertex w and new edges wx, wy and wz. Terminals are distinguished vertices that cannot be deleted by reductions and transformations. When a specified subset A of the nodes is distinguished as terminal nodes, we require that for any a A, a cannot take the place of the degree one or degree two vertex in the reductions R1 and R2, or the degree three vertex in the Y transformation, described above. Working with terminals introduces the question of deciding what should be considered as an adequate list of irreducible configurations. In [5], Feo and Provan introduce an operation which we call an FP-assignment: it reassigns a degree one terminal vertex a incident to a non-terminal vertex b. This is done by eliminating the vertex a and then changing the status of b by considering b a terminal vertex. FP assignment(see Figure 1). Eliminate a terminal vertex a of degree one that is the end of a pendant edge and then change the status of the other end vertex of the edge, say b, by now considering b as a terminal vertex. Terminal vertexa b Terminal vertexb Figure 1: FP-assignment This transformation implicitly re-embeds terminal pendant edges, but it remains consistent by keeping track of these reassignments. In other words, when we apply this transformation we forget how some pendant edges where originally embedded in order to be able to perform further reductions. In applications one must keep track of the embedding of pendant edges in order to reverse correctly the reduction process. By allowing FP-assignments the set of irreducible configurations is decreased substantially without changing the nature of the reduction problem (particularly useful in applications and reduction algorithms), with an unnatural long list of irreducible configurations as Figure 2 shows. terminal vertex non terminal vertex Figure 2: Artificial irreducible 3-terminal plane graphs Definition 1.1 A connected plane graph is called terminal Y reducible if it can be reduced to eliminate all non-terminal vertices by using R0, R1, R2, R3, Y, Y and FP-assignments. 2

3 Contracting an edge e = uv, consists of deleting e and identifying its two endpoints u = v to make a single vertex. A minor of G is a graph formed by a sequence of edge deletions, edge contractions and deletion of isolated vertices. Combining the previous two concepts, we can have a minor H of a graph G with terminals. Specifically, a terminal minor is formed using the same three minor operations as above, except that we forbid contracting an edge joining two terminals and deleting an isolated terminal. It follows that H has the same number of terminals as G. In 1966 G.V. Epifanov proved the following important theorem : Theorem 1.1 (Epifanov [4]). Each connected plane graph with two terminals is Y reducible to a single edge between the terminals. This theorem has the following corollary: Corollary 1.2 (Epifanov [4]), Grünbaum [8]). Each connected plane graph is Y reducible to a vertex. Since Epifanov s and Grünbaum s work simpler proofs have been found (see [5] and [12]). For generalizations see for instance [11]. It is important to notice that not every connected graph is Y reducible: one such graph is the complete bipartite graph K 4,4. Theorem 1.3 (Truemper [12], Gitler [6] and Archdeacon,etal [2]). Suppose that H is a terminal minor of G. If G is terminal Y reducible, then H is terminal Y reducible. It follows that if a connected graph G is Y reducible, then each connected minor H of G is Y reducible as well. Theorem 1.4 (Gitler [6]). A two connected plane graph with three terminals is Y reducible to K 3, where the vertices are the original three terminals. We first give an outline of the proof given by Gitler [6]. Let G be a two connected 3-terminal plane graph. The first step is to show that G can always be embedded on some grid F which is also a 3-terminal plane graph, and has G as a minor. Then, by using the corner and modified corner algorithms given in [6], the second step is to show that F is Y reducible to a specific 3-terminal graph, called a perfect mirror M. The third step consists in showing that M is Y reducible to K 3. Finally, the minor G of F is Y reducible to K 3 by Theorem 1.3. Connected plane graphs with more than three terminals are in general not reducible ([2, 6]), see the example in Figure 3. = terminal = terminal Four terminal graph G Medial graph of G Figure 3: Non reducible 4 terminal plane graph 3

4 2 Preliminaries Given a connected plane graph G, its medial graph M(G) is defined as follows. The vertices of M(G) are the edges of G. Each face f = e 1,..., e r of length r in G determines r edges {e i e i+1 : 1 i r 1} {e r e 1 } of M(G). In this definition, a loop e that bounds a face is viewed as a face of length one, and so determines one edge of M(G), which is a loop on e (similarly for pendant edges). The graph M(G) is four regular and plane. Let G denote the dual of the graph G, then M(G) M(G ); G and G are called the face graphs of M(G). Any connected four regular plane graph is the medial graph of some pair of dual plane graphs. When speaking of the medial graph M(G) we will always take as reference the face graph G that does not contain a vertex corresponding to the infinite face in M(G), and we refer to it as the black graph of M(G). The faces in M(G) corresponding to the vertices in G are called black faces and the others white faces. Let M be a 4-regular graph embedded on the plane. The straight decomposition K(M) of M is the decomposition of the edges of M into closed curves C 1,..., C k, (called closed geodesic arcs) in such a way that, each edge is traversed exactly once for these curves and in each vertex v of M if e 1, e 2, e 3 and e 4 are the edges incident to v in cyclic order, then e 1 ve 3 are traversed consecutively (in one way or the other); in this case e 1 has as direct extension e 3 (e 3 has as direct extension e 1 ). Similarly e 2 ve 4 are traversed consecutively (in one way or the other). The straight decomposition is unique up to choice of the beginning vertex of the curves, up to reversing the curves and up to permuting the indexes of C 1,..., C k. We always view a given plane connected graph G with k-terminal vertices, through its medial graph as a black graph, together with its straight decomposition K(M(G)) into closed geodesic arcs and we call the faces in M(G) corresponding to the k terminal vertices of G, the terminal faces of M(G). 2.1 Y operations on the medial graph The Y operations introduced in section 2, have a direct translation on to M(G) as four medial reductions and two medial transformations (see [3]). If a Y operation O (transformation or reduction) is applied on G giving O(G) then the medial graph of the resulting graph M(O(G)) is M(G) after the application of the corresponding medial operation O M. In other words M(O(G)) O M (M(G)). In general we denote a medial operation with its Y name followed by M as subscript. The ( Y) M operations are defined as (see Figure 4): R0 M Medial loop reduction. Topologically collapse a white loop (a loop enclosing a white face) in M(G) to a single vertex of degree two and then omit this vertex. R1 M Medial degree-one reduction. Topologically collapse a black loop (a loop enclosing a black face) in M(G) to a single vertex of degree two and then omit this vertex. R2 M Medial series reduction. Topologically collapse a digon enclosing a black face in M(G), to a single vertex of degree four, thus identifying the end vertices of the digon. R3 M Medial parallel reduction. Topologically collapse a digon enclosing a white face in M(G), to a single vertex of degree four, thus identifying the end vertices of the digon. Each of these reductions decreases the number of vertices, edges or faces in M(G). transformations on M(G) are: The other two (Y ) M Medial Y transformation. Topologically collapse a black triangle face to a single vertex of degree six, thus identifying the vertices of the triangle, then expand this vertex to a white triangle whose edges are incident to the black regions. ( Y) M Medial Y transformation. Topologically collapse a white triangle face to a single vertex of degree six, thus identifying the vertices of the triangle, then expand this vertex to a black triangle whose edges are incident to the white regions. 4

5 collapse expand omit expand collapse collapse expand expand collapse collapse expand expand collapse Figure 4: Medial ( Y) M transformations As before, when a specified subset F of black faces is distinguished as terminal faces, we require that for any f F, f cannot take the place of the black loop or the black face in the medial operations R1 M and R2 M, or the black triangle in the medial operation (Y ) M, described above. 5

6 We now give the formulation of the FP-assignment on the medial graph: FP M Medial FP assignment (see Figure 5). Eliminate a black terminal loop a provided that its cone b (the cone of a loop (or 1 lens) is defined in Section 3) corresponds to a non-terminal face, henceforth b is considered a terminal face. Terminal face a Terminal faceb Figure 5: FP-assignment If a medial graph with terminal faces can be reduced by using the operations defined above, to eliminate all non-terminal black faces then it is called terminal ( Y) M reducible. 3 k-lenses Throughout the remaining part of this paper G will denote a four regular, connected, plane graph (a medial graph), together with its straight decomposition K(G). We follow [8]. A path v 0 v 1... v n in G is a geodesic arc if and only if v i 1 v i has v i v i+1 as direct extension for 1 i < n; for a closed geodesic arc, v 0 = v n, and v n 1 v n has v 0 v 1 as direct extension. We call simple paths or simple geodesic arcs, those paths or geodesic arcs which do not have self intersections. Given an integer k > 0, a subgraph L of G is called a k-lens provided: L1: L consists of a simple closed path (see Figure 6.a) R = v 0,0 v 0,1... v 0,i0 v 1,0 v 1,1... v 1,i1... v k 1,0 v k 1,1... v k 1,ik 1 v 0,0 called the boundary of L, and all the vertices and edges of G contained in one of the connected components of the complement of R in the 2-sphere. We call interior of L to such a connected component and all the vertices and edges lying in it are called the inner vertices and edges of R. L2: For 0 j < k v j,0 v j,1... v j,ij v (j+1) mod k,0 are simple geodesic arcs, called geodesic boundary arcs. No inner edge of R is incident to the vertices v 0,0, v 1,0,..., v k 1,0, which are called the poles of the k-lens. Figure 6 gives examples of k-lenses. a) a 4-lens, b) and c) two 2-lenses, d) a configuration that is not a 2-lens and e) and f) two 1-lenses. In a generic way, a lens is a k-lens for some value of k. Let L be a lens, it is singular, if it does not contain any inner vertex or edge; otherwise it is non singular. A chord of L is a simple geodesic arc P = p 0 p 1... p k such that p 0 and p k belong to the boundary of L but the other vertices and edges in P are interior to L. The rays of a pole w in L are the edges in G which are incident to w but do not belong to the boundary of L. By definition, the rays of a pole in L are not in the interior of L. Given a pole w of a lens L there is a unique face F interior to L incident to w. Of the remaining faces incident to w exactly one, say H, shares color with F (since the boundary of a lens is a Jordan curve, F and H are distinct faces). We call H the cone associated to the pole w of L. When we talk about the rays and cone of a 1-lens we do not make an explicit reference to the pole. 6

7 V2,0 b) V2,1 V1,3 V3,0 V1,1 V1,2 V1,0 e) V0,2 V0,1 V 0,0 c) a) d) f) Figure 6: Lenses examples A k-lens (k 2) L in G is indecomposable if and only if it does not contain properly a 1-lens or a 2-lens. A 1-lens is indecomposable, if and only if, it does not contain properly in its interior a 1-lens or a 2-lens (see Figure 7). A k-lens which is not indecomposable is decomposable. All the lenses in Figure 6 are indecomposable but the 1-lens in Figure 12.b is not. The following result is an immediate consequence of this definition. Lemma 3.1 A k-lens is indecomposable if and only if the following conditions are true: I1: Every inner edge in L belongs to a chord of L. I2: Two different chords in L meet in at most one vertex. I3: If k > 1 then each chord in L intersects the boundary of L in exactly two vertices: each of them in a different geodesic boundary arc. decomposable 1 lens indecomposable 1 lens decomposable 2 lens indecomposable 2 lens Figure 7: Decomposable and indecomposable k-lens (k = 1, 2) We say that an indecomposable k-lens (k 2) can be emptied, if after a finite sequence of (Y ) M or ( Y) M transformations it becomes singular. Note that after applying any of these transformations to an indecomposable k lens the resulting k lens is indecomposable. The following result is used several times in our proofs. 7

8 Lemma 3.2 For every G, any indecomposable k-lens L (with k = 2 or 3) can be emptied through a finite sequence of (Y ) M or ( Y) M operations. Proof: The proof for k = 2 was given by Grünbaum [8]. Now assume that L is a 3-lens. We first prove the following two elementary results: Proposition 3.3 Let L be an indecomposable 3-lens with geodesic boundary arcs A, B and C. Suppose L does not contain chords meeting both A and B, then there exists a triangular face R in L sharing an edge with C (see Figure 8). A B C R Figure 8: Illustration of Lemma 3.3 Proof: By induction on the number of chords in L: when this number is zero, L itself plays the role of R. If we suppose the property true for n chords then there exists a singular triangle R in L sharing an edge with C; when we add in a new chord, one of the following cases must hold: R C Figure 9: The new chord meets R i) The new chord does not meet R. In this case R plays the role of R and the result is true. ii) The new chord meets R (see Figure 9). No other case is possible given that L is indecomposable. Now, since one of the resulting sub-regions of R is a triangular face having an edge on C, hence the result follows. Corollary 3.4 Any indecomposable 3-lens L with boundary arcs A, B and C which does not contain chords meeting simultaneously A and B can be emptied through medial (Y ) M or ( Y) M operations. 8

9 Proof: The triangular face R determined by Lemma 3.3 can be eliminated from L by applying a (Y ) M or a ( Y) M operation. The resulting 3-lens satisfies the conditions in the hypothesis of Proposition 3.3 and, again, there exists a triangular face R which can be eliminated. We may continue in this way until the original 3-lens is emptied. Now, to proof the Lemma, let A, B and C be the boundary arcs of L. Among the chords in L meeting simultaneously A an B we choose one (in general there are several possibilities), named D, such that in the 3-lens with boundary arcs A, B and D no chord meets simultaneously A and B. We empty this 3-lens using the method in Corollary 3.4 and then we eliminate D from the interior of the lens by a (Y ) M or ( Y) M transformation. We continue in this way until the original 3-lens becomes singular. This completes the proof. Figure 10: After a medial series or parallel reduction a 1-lens becomes a 2-lens The last Lemma is not valid for k = 1: in Figure 10 we show a 1-lens to which only a medial series or parallel reductions can be applied, notice that after applying any of these reductions the 1-lens is transformed into a 2-lens and the structure of the original 1-lens is lost. So, emptied does not apply in this context. Figure 11: A 4-lens which cannot be emptied by ( Y) M transformations The Lemma does not hold for k > 3. Figure 11 depicts a 4-lens which cannot be emptied by the application of any ( Y) M transformations. 4 One-step ( Y) M reducibility We study how to obtain a reduction in a medial graph with terminal faces by ( Y) M operations. Definition 4.1 A medial graph G is one-step ( Y) M reducible, if it admits a sequence of ( Y) M operations where all of them are ( Y) M or (Y ) M except the last one, which is a reduction of type R0 M, R1 M, R2 M, R3 M, or F P M. A t-scheme in a graph G is a 1-lens T containing a chord P. Let S = w 0... w j w 0 and P = p 0... p i be the boundary and the chord of T, respectively. Assume that p 0 = w s and p i = w r (0 < r < s j). The 2-lens 9

10 of T, denoted L 2 (T ), is the 2-lens with boundary p 0... p i 1 w r w r+1... w s 1 p 0. The 3-lens of T, denoted L 3 (T ) is the 3-lens with boundary w 0... w r 1 p i... p 1 w s... w 0. (See Figure 12). wj 2 L (T) p 0 p1 3 L (T) p i w2 w 1 w 0 cone a) b) Figure 12: (a) A t-scheme T. (b) A t-scheme in a four regular plane graph Lemma 4.1 Let L be an arbitrary k-lens in a graph G with k = 1 or k = 2. Then it must contain a singular 1-lens or an indecomposable 2-lens. Proof: We start with k = 2. If L is indecomposable the result is true, otherwise it contains a 1-lens or a 2-lens. If L contains a 1-lens, say M, we have two possibilities: L.1 M is singular and the result is true. L.2 M is not singular. Now, if M is not a t-scheme then it contains a geodesic arc which intersects itself in its interior, so M contains a 1-lens T which we take in place of M, and we repeat this replacement until an empty loop or a t-scheme is found. If M is a t-scheme then consider L 2 (M) instead of L (we mean, L has been replaced now by L 2 (M)). If the new L is indecomposable the proof is over, otherwise we start again the analysis in a recursive way. Since G is finite we will finish after a finite number of steps. If L does not contain a 1-lens then, since L is decomposable, it must contain a 2-lens N which takes the place of L and we continue, as before, in a recursive way. This completes the analysis for k = 2. For k = 1 the analysis corresponds to Cases L.1 and L.2 given before. This result has the following corollary. Corollary 4.2 Let L be an arbitrary terminal free k-lens in a graph G, with k = 1 or k = 2. Then G is one-step ( Y) M reducible. Proof: By Lemma 4.1 L contains a singular 1-lens or an indecomposable 2-lens. In the former case G reduces its number of edges and faces after the application of a R0 M or a R1 M operation. In the latter case the indecomposable 2-lens can be emptied by Lemma 3.2, and G reduces the sum of the number of vertices, edges and faces, after the application of a R2 M or a R3 M operation. For 3-lenses we have a similar corollary: Corollary 4.3 Let L be an arbitrary terminal free 3-lens in a graph G. If L is not indecomposable then G is one-step ( Y) M reducible. 10

11 Proof: Since L is not indecomposable it contains a 1-lens or a 2-lens. The result follows from Corollary 4.2. Lemma 4.4 If G has a non singular 1-lens T containing at most one terminal face then G is one-step ( Y) M reducible. Proof: Suppose that T is not a t-scheme, hence T has no chords, and thus it contains at least a 1-lens T. If T is singular and does not contain a terminal face then the result follows because we can apply a R0 M or a R1 M operation. If T is singular and contains a terminal face then the cone of T is not a terminal face and so we can apply the F P M reduction and the result is true. If T is not singular and terminal free then the result follows from Corollary 4.2. If T is not singular and contains at most one terminal face then T is replaced with T and we start the analysis in a recursive way. Now, if T is a t-scheme then one of L 2 (T ) or L 3 (T ) is terminal free. In the first case the result follows from Lemma 3.2. In the second case, if L 3 (T ) is not indecomposable then the result is true by Corollary 4.3. If L 3 (T ) is indecomposable then we empty it using the method given in the proof of Corollary 3.4 (in order to simplify the explanation, we still denote by T the resulting 1-lens). After this L 3 (T ) becomes a triangular non terminal face which can be eliminated from T by the application of a ( Y) M or (Y ) M operation. After this operation the cone of T is not a terminal face and we have two possible situations: 1. The 1-lens T is a loop containing a terminal face. Then its cone does not contain a terminal face and we can apply an F P M reduction. 2. The 1-lens T is not singular and contains at most one terminal face. In this case we start again with the analysis on T in a recursive way. Corollary 4.5 If G has a 1-lens T containing at most one terminal face, and the cone of T is terminal free then G is one-step ( Y) M reducible. Proof: If T is not singular then the result follows from Lemma 4.4. Otherwise T satisfies one of the following conditions: 1. T is terminal free, so the result is true since we can apply one of the reductions R0 or R1. 2. T contains exactly one terminal face. The result is true again because we can apply the F P M reduction. 5 Main Theorem We prove that any graph with three terminal black faces is ( Y) M reducible to one of the graphs M(P 3 ) or M(P 3 ) given in in Figure 13. We call these graphs the irreducible ( Y) M graphs or simply irreducible graphs. In order to reduce a graph with three terminal black faces, we first show (Theorem 5.1) that any graph not isomorphic to the irreducible graphs must contain at least one of four configurations. Next we show that each of these four configurations is one-step ( Y) M reducible (Theorem 5.2). For this we use the results of sections 3 and 4 to show that there always exist a terminal free indecomposable k lens (k = 1, 2) or a non singular 1 lens containing at most one terminal face, or a singular 1 lens containg a terminal face with terminal free cone. Since each of these is one step ( Y) M reducible, a reduction is obtained. Hence 11

12 (a) (b) (c) (d) Figure 13: Irreducible graphs: a) P 3, b) P 3, c) M(P 3 ) and d) M(P 3) the sum of the number of vertices, edges and faces is decreased. Then the main result (Theorem 5.7) follows by successive application of a finite sequence of one-step medial reductions. Henceforth G 3 will denote a connected four regular plane graph with three terminal black faces, together with its straight decomposition. Let C be a closed geodesic arc in G 3. A vertex x in C is called a selfintersection vertex of C, if all the edges which are incident to x belong to C. The self-intersection number of C is the number of self-intersection vertices in C. Theorem 5.1 Let G 3 be a graph not isomorphic to one of the irreducible graphs, then G 3 (see Figure 14): C1 contains a closed geodesic arc C with self-intersection number greater or equal than two and such that there exists a non singular 1-lens whose boundary is contained in C, or C2 contains a closed geodesic arc C with self-intersection number one and the other closed geodesic arcs with self-intersection number less than or equal to one, or C3 it has at least one 1 lens and all its 1 lenses are singular, or C4 all its closed geodesic arcs have self-intersection number zero. C1 C2 C3 C4 Figure 14: Configurations in Theorem 5.1 Proof: Let C be a closed geodesic arc in G 3 with at least one non singular 1 lens and maximum selfintersection number n. If one such C does not exist, then let C be a closed geodesic arc with maximum self-intersection number n. Now, if n 2 and C contains a non singular 1-lens then C1 is satisfied, otherwise the following must hold: 12

13 1. (n = 0): No closed geodesic arc has self-intersection vertices and C4 is true. 2. (n = 1): C2 is satisfied since n is maximum, and one of the two 1 lenses in C is non singular (G 3 is connected and has more than three faces). 3. (n 2): All the 1-lenses in G 3 are singular, this is condition C3. Theorem 5.2 Let G 3 be a graph not isomorphic to one of the irreducible graphs, then G 3 is one-step ( Y) M reducible. Proof: We prove that each one of the configurations given in Theorem 5.1 is one-step ( Y) M reducible. This is proved, respectively, for the Cases C1, C2, C3 and C4 in Propositions 5.3, 5.4, 5.5 and 5.6 below: The key idea in the proof of the following Propositions is to determine (with k 3): four k-lenses or three k-lenses and a cone of one of these lenses or two k-lenses and their cones; whose interiors are pairwise disjoint. One of these lenses or cones must be terminal free and after some ( Y) M operations we reduce it. If a 1-lens contains at most one terminal face and its cone is terminal free, then we do an F P M reduction if necessary. Proposition 5.3 Let G 3 be a graph containing a closed geodesic arc C with self-intersection number greater or equal than two and such that there exists a non singular 1-lens T whose boundary is contained in C. Then G 3 is one-step ( Y) M reducible. Proof: Let S = w 0 w 1... w j w 0 be the boundary of T and let S = w 1... w j w 0 x 1... x k be the geodesic arc that contains w 1... w j w 0 in such a way that x k is the first self-intersection in S. The geodesic arc S is a local direct extension of S at w j. The value of k is well defined because, by hypothesis, the self-intersection number of C is at least two. The geodesic arc S locally looks like one of the configurations in Figure 15. w j w 0 T N T T N x 1 x k w1 L x k N w 0 L a) b) w0 c) x k N N T T T x k w0 N w 0 x k w 0 x k d) e) f) We distinguish two possibilities: Figure 15: Possible local direct extensions of w 1... w j w 0 at w j 13

14 x k {w 1... w j } (see incises a and b in Figure 15). In this case there exists a value r (1 r j) such that w r = x k. The closed geodesic arc w r w r+1... w j w 0 x 1... x k is the boundary of a 1-lens (N ) and w 0... w r x k 1... x 1 w 0 is the boundary of a 2-lens (L). The interior of N, L and T are mutually disjoint. If one of these lenses is terminal free then the Proposition follows from Corollary 4.2. The only non trivial case is when both of L and N contain terminals, then T contains at most one terminal face and the Proposition follows from Lemma 4.4. x k {x 1,..., x k 1 } (see incises c, d, e and f in Figure 15). In this case there exists a value m (1 m < k) such that x m = x k, and thus x m x m+1... x k is the boundary of a 1-lens N such that the interior of T and N are disjoint. If these regions were not disjoint then we would have the following possibilities: 1. The boundary of T intersects the boundary of N. Thus the first self-intersection in S should appear in x j = w n for appropriate values of j and n (j < m), but this contradicts the fact that x k = x m is the first self intersection. 2. The interior of T is contained in the interior of N. Since the rays of N are in its exterior then the geodesic arc S should cross the boundary of N (just to meet one of the rays of T ), in some vertex x o with o < k but this contradicts the selection of k. 3. The interior of N is contained in the interior of T. This is impossible by an argument similar to the previous one. We conclude that the interior of T and N are disjoint. Now, if one of the lenses T or N is terminal free then the proposition is true by Corollary 4.2. If T contains at most one terminal face then the result follows from Lemma 4.4. The same thing happens when N is not singular and contains at most one terminal face. If N is a loop containing a terminal face but its cone is not a terminal face then we apply the F P M reduction and the result is true again. Any other situation is covered by these cases. Proposition 5.4 Let G 3 be a graph containing a closed geodesic arc C with self-intersection number one. Assume that the other closed geodesic arcs in G 3 have self-intersection number less than or equal to one (the other case is contemplated in Proposition 5.3). Then G 3 is one-step ( Y) M reducible. Proof: Let us assume that C = c 0 c 1... c k c 0 c k+1 c k+2... c l c 0 where c 0, c 1, c 2,..., c l are pairwise distinct. This closed geodesic arc forms two 1-lenses in G 3 and one of them, say T = c 0 c 1... c k 1 c 0 is not singular, otherwise G 3 would contain at most two black faces which contradicts the fact that G 3 contains at least three terminal black faces. Thus there exists a geodesic arc D which intersects the geodesic boundary arc of T in some vertex, say c j (1 j k). We can assume that D = c j d 1 d 2... d m c n... where c j d 1 is an edge in the interior of T (that is we choose the direction towards the interior of T to describe the geodesic arc D) and c n is the second vertex in c j d 1 d 2... d m c n... belonging to the boundary of T (it must exist because the boundary of T is a Jordan curve) and we can assume too that j < n. Finally, we can also assume that if D contains a self-intersection vertex inside one of the two 1-lenses of C then it belongs to the path c j d 1 d 2... d m c n. Now we construct all the possible local direct extensions of D from c j that meet C in at most four vertices. We determine in the embedding consisting of such extension of D union C, one of the configurations D1, D2, D3 or D4 defined bellow. Then we prove that each one of them is one-step ( Y) M reducible. D1 Configuration with one 1-lens L 1 with cone x and two k-lenses L 2 and L 3 (with k {1, 2}) in such a way that all these lenses and cone are mutually disjoint. (See incises a), b), d) and g) in Figure 16.) D2 Configuration with two 1-lenses L 1 and L 2 with cones x 1 and x 2 respectively, in such a way that all these lenses and cones are mutually disjoint. (See incise c) in Figure 16.) 14

15 L 1 L 2 x L 2 L 3 L 1 x 1 x 2 L 1 L 2 x L 3 L 1 L 2 L 3 a) b) c) d) L 2 L 2 L 1 x L 3 L 1 L 2 L 1 e) f) g) Figure 16: Some of the configurations for the proof of proposition 5.4 D3 Configuration with one t-scheme L 1 and one 1-lenses L 2 with cone x in such a way that all these lenses and cone are mutually disjoint. (See incise e) in Figure 16.) D4 Configuration with two t-schemes L 1 and L 2 mutually disjoint. (See incise f) in Figure 16.) Let T be the 1-lens c 0 c k+1... c l c 0. We have the following cases: Case 1 The geodesic arc c j d 1 d 2... d m c n contains a self-intersection vertex of D. Then d r = d s for 1 r < s m and the other vertices in this path are pairwise distinct. Case 1.1 The cone of the 1-lens d r d r+1... d s is different from the cone of T, hence we obtain the structure given by configuration D2. Case 1.2 The cone of the 1-lens d r d r+1... d s coincides with the cone of the 1-lens T. We take the local direct extension D = c j d 1 d 2... d m c n e 1... e p... where e p is the first vertex belonging to C in the geodesic arc e 1... e p... (it does exist because in the worst case e p = c j.) Case The vertex e p belongs to T, so e p = c o for some 1 o k. Case o < n. Then L 1 = d r d r+1... d s, L 2 = T and L 3 = c o c o+1... c n e 1 e 2... e p 1 c o (the 2-lens with poles c o and c n ), hence we obtain the structure given by configuration D1. Case o > n. Then L 1 = d r d r+1... d s, L 2 = T and L 3 = c o c o 1... c n e 1...e p 1 c 0 (the 2-lens with poles c o and c n ), hence we obtain the structure given by configuration D1. Case The vertex e p belongs to T, so e p = c o for some k + 1 o l. We take the local direct extension D = c j d 1 d 2... d m c n e 1... e p... e r... where e r is the second vertex belonging to T in the geodesic arc e 1... e p... e r. Then L 1 = T and L 2 = d r d r1... d s obtaining the structure given by configuration D3. Case 2 The geodesic arc c j d 1 d 2... d m c n does not contain a self-intersection vertex of D. Take the local direct extension D = c j d 1 d 2... d m c n e 1... e p... where e p is the duplicated vertex in c j d 1 d 2... d m c n e 1... e p... or e p is the first vertex in C appearing in e 1... e p.... Then: 15

16 Case 2.1 The vertex e p is the first duplicated vertex in c j d 1 d 2... d m c n e 1... e p.... Then e p must be equal to e u (1 u < p) (because D does not have self-intersection vertices in the interior of the 1-lenses of C) and so L 1 = e u... e p, L 2 = T and L 3 = T obtaining the structure given by configuration D1. Case 2.2 The vertex e p is the first one in C appearing in e 1... e p.... Case The vertex e p belongs to the boundary of T. Then e p = c o (1 o k). Case o < n. Then L 1 = T, L 2 = c j d 1 d 2... d m c n c n 1 c n 2... c j (the 2-lens with poles c j and c n ), and L 3 = c o c o+1... c n e 1... e p 1 c 0 (the 2-lens with poles c 0 and c n ) obtaining the structure given by configuration D1. Case o > n. Then L 1 = T, L 2 = c j d 1 d 2... d m c n c n 1 c n 2... c j (the 2-lens with poles c j and c n ), and L 3 = c o c o 1... c n e 1 e 2... e n 1 c o (the 2-lens with poles c 0 and c n ) obtaining the structure given by configuration D1. Case The vertex e p belongs to the boundary of T. Then e p = c o with k + 1 o l. Take the local direct extension D = c j d 1 d 2... d m c n e 1... e p... e r... where e r is the second vertex in the boundary of T in e 1... e p... e r.... We have that T and T are both t-schemes and we obtain the structure given by configuration D4. We prove that any graph G 3 containing at least one of the configurations D1, D2, D3 and D4 is one-step ( Y) M reducible. D1: if one of the lenses L 1, L 2 or L 3 is terminal free then the result is true by corollary 4.2, otherwise the cone x is empty and the result is true by Corollary 4.5. D2: without loss of generality we can assume that L 1 contains a number n of terminal faces which is greater or equal than the corresponding number in L 2. If n 2 or n = 1 and the cone of L 1 is a terminal face, then the result is true because L 2 is terminal free and we can apply Corollary 4.2 or the cone of L 2 is terminal free and L 2 contains at most one terminal face and we can apply Corollary 4.5. If n = 1 and the cone of L 1 is terminal free then the result follows from Corollary 4.5. If n = 0 then the result is true because we can apply Corollary 4.2. D3: if the t-scheme L 1 contains more than one terminal face then L 2 is terminal free and we can apply Corollary 4.2 or the cone of L 2 is terminal free and L 2 contains at most one terminal face and we can apply Corollary 4.5. If L 1 contains at most one terminal face then the result follows from Lemma 4.4. D4: necessarily one of the t-schemes L 1 or L 2 contains at most one terminal face, the result is true because we can apply on such lens Lemma 4.4. Proposition 5.5 Let G 3 be a graph with at least one 1 lens, where all 1-lenses are singular, and which is not isomorphic to the irreducible graphs. Then G 3 is one-step ( Y) M reducible. Proof: In the graph G 3 all the 1-lenses are singular and thus no closed geodesic arc in G 3 has self-intersection number one (if such a geodesic arc existed, say B, then since G is connected there would exist another geodesic arc which meets B and one of the two 1 lenses of B would be non singular). Configuration E1. If G 3 has more than three 1-lenses then one of them is terminal free. A similar argument to the one given in the final part of the proof of Proposition 5.3 shows that the interior of all the 1-lenses in G 3 are mutually disjoint, thus G 3 is one-step ( Y) M reducible. Configuration E2. If G 3 contains three 1-lenses then one of them is terminal free or each one contains a terminal face but their cones are terminal free (a 1-lens cannot be the cone of some 1-lens because no closed geodesic arc in G 3 has self-intersection number one). We reduce the graph after the application of a R0 M, a R1 M or a F P M reduction. In this case the proposition is true. Configuration E3. If G 3 contains exactly two 1-lenses whose cones are in different faces then one of the two 1-lenses is terminal free or one of the cones is terminal free and the proposition is true. 16

17 We only need to analyze the case in which G 3 contains exactly two 1-lenses S and T whose cones are in a common face. Both 1-lenses should belong to the same closed geodesic arc C (since the minimum number of 1-lenses in a closed geodesic arc with self-intersection number greater than zero is two, if S and T are in different closed geodesic arcs then the number of 1-lenses in G 3 is at least four but this is not possible) and in fact C is the unique closed geodesic arc in G, so the rays of S should be connected with the rays of T through simple geodesic arcs. Let us assume that S = s 0 s 1... s n s 0 and T = t 0 t 1... t l t 0. Let P = s 0 p 1... p m t 0 be the simple geodesic arc from s 0 to t 0 with maximun number of vertices. If m 2 then there is a geodesic arc Q = p r q 1... q n p s, r s such that {q 1... q n } {s 0 s 1... s n t 0 t 1... t l p 1... p m } = φ. If p r p r+1... p s q n q n 1... q 1 p r is a 2 lens L, then its interior does not contain S nor T. Hence, we have Configuration E4. If one of S, T and L is terminal free or S contains exactly one terminal but its cone is terminal free. Then G 3 is one-step ( Y) M reducible (Corollaries 4.2 and 4.5). Otherwise p r p r+1... p s q n q n 1... q 1 p r is a Jordan curve containing in one of the regions it defines, the cone of S and in the other the cone of T, but this is a contradiction. If m = 1 the only case is the one depicted in figure 17.a, which is a contradiction since we assumed it is not isomorphic to M(P 3). If m = 0, then we have one of the two cases depicted in figure 17.b and 17.c. The configuration in figure 17.b is not possible since we assumed that G 3 was not isomorphic to M(P 3 ). The configuration depicted in figure 17.c is not posible since we assumed that G 3 contains at least three black faces. So, G 3 is one-step ( Y) M reducible. a) b) c) Figure 17: Illustration of Proposition 5.5 Proposition 5.6 Let G 3 be a graph in which all closed geodesic arcs have self-intersection number zero. Then G 3 is one-step ( Y) M reducible. Proof: We can assume that G 3 contains more than two closed geodesic arcs (otherwise we cannot have three terminal faces). Let C = c 0 c 1... c m c 0 and D = d 0 d 1... d m d 0 be two intersecting closed geodesic arcs in G 3. Without loss of generality we can assume that c 0 = d 0 and that the edge d 0 d 1 is in the interior of C. Now we prove that it is always possible to find a local direct extension of D from d 0 in such a way that the union of such an extension with C contains four mutually disjoint 2-lenses. Since G 3 has only three terminal faces one of the four 2-lenses must be terminal free and the result will follow from Corollary 4.2. If C and D meet in exactly two vertices (see incise a) in Figure 18) then there exists a vertex c i = d j C D(1 < i m, 1 < j n). The four 2-lenses d 0... d j c i 1... d 0, d 0... d j c i+1... c 0, c 0... c i d j+1... c 0 and c 0 c m c m 1... c i+1 c i d j+1 d j+2... d n c 0 are mutually disjoint. Otherwise D = d 0 d 1... d j... d l... d n d 0 in such a way that c i = d j and c k = d l are, respectively, the second and third vertex of C in d 0 d 1... d k... d l... d n. We have two possibilities: i < k (see incise b) in Figure 18) or i > k (see incise c) in Figure 18.) If i < k then the four 2-lenses c 0 c 1... c i d j 1... d 1 c 0, d 0 d 1... d j c i+1... c m d 0, c i c i+1... c k d l 1 d l 2... d j+1 c i, and c i c i 1... c 0 c m c m 1... c k d l 1 d l 2... c i are mutually disjoint. On the other hand, if i > k then the four 2-lenses c 0 c 1... c i d j 1... d 1 c 0, d 0 d 1... d j c i+1... c m d 0, c k c k+1... c i d j+1... d l 1 c k and c i c i+1... c m c 0 c 1... c k d l 1 d l 2... d j+1 c i are mutually disjoint. This ends the proof of Theorem

18 a) b) c) Figure 18: Illustration of Proposition 5.6 Theorem 5.7 Any graph G 3 can be ( Y) M reduced to one of the irreducible graphs. Proof: Since any graph G 3 which is not isomorphic to one of the irreducible graphs is one-step ( Y) M reducible and in each one-step reduction the sum of the number of vertices, edges and faces is decreased, after a finite number of one-step ( Y) M reductions we reach one of the irreducible graphs. It readily follows that any connected planar graph with at most three terminals is ( Y) reducible to a vertex, an edge, a path P 3 or the graph P 3, where the vertices are the terminal vertices of the graph. It is worth noticing that the proof implicitly gives an algorithm for reducibility that can be implemented efficiently [7]. We build a data structure that contains a representation of the indecomposable 1-lens and 2-lens. For each lens we take the number of vertices on its boundary as a measure of its complexity, we also save the number, if any, of the terminals it contains. Then we iteratively proceed as follows: we find a lens of minimum complexity that is one-step ( Y) M reducible and empty it. By theorems 5.1 and 5.2 we know such lens always exists. Since in each one-step ( Y) M reduction the sum of the number of vertices, edges and faces is decreased, after a finite number of iterations we reach one of the irreducible graphs. More explicitly: a direct implementation of the constructive approach in the proof of Theorem 5.1 produces the following algorithm. Algorithm 5.1 [To ( Y) M reduce a graph G 3 ] Input: The graph G 3. Output: The ( Y) M reduction of G 3. Method: 1. G G 3 2. While G is not ( Y) M reduced do 3. Determine the closed geodesic arc pattern P in the statement of theorem 5.1 contained in G. In a precise way the possibilities are: patterns c1.a,... c1.f in Proposition 5.3, patterns D1, D2, D3 and D4 in proposition 5.4, patterns E1, E2, E3 and E4 in proposition 5.5 and patterns c4.a, c4.b, c4.c in proposition Identify the 1- and 2-lenses in P. 5. Classify each face f in G in terms of the 1 or 2-lenses in P containing f. Identify the 1-lenses cones in P. Finally determine the lens L in P that should be emptied. 6. If L is a t-scheme containing a terminal face then 7. Find recursively the singular terminal free 1 -lens, or the indecomposable terminal free 2-lens or the 1-lens with exactly one terminal face and empty cone inside L to make a 1-step ( Y) M reduction (see Lemma 4.4 and Corollary 4.5). 18

19 8. If L is a 1-lens containing exactly one terminal face and the cone of L is terminal free then 9. Apply the medial FP assignment to L. 10. else Empty and reduce the lens L (see Corollary 4.2). 11. Replace G by the reduced graph. 12. Return G. Proposition 5.8 Algorithm 5.1 runs in O( V (G 3 ) 4 ) worst time. Proof: In this proof we denote by n i = V (G 3 ) i + 1 the number of vertices in G in iteration i at step 2 in the algorithm. We denote by T j (n i ) the time complexity of step j to accomplish iteration i. Steps 3 and 4 require traversing the edges in G along the geodesic curves. In the route every edge should be oriented in the traversing direction; then, for each geodesic we should count the number of selfintersections and look at the relative orientation of the geodesic in these vertices to determine the pattern P as well as the 1- and 2-lenses it contains. Since in the worst case every edge in G is reached and the number of edges in G is proportional to n i, this process is completed in time T 3 (n i ) + T 4 (n i ) = O(n i ) The identification of the 1- or 2-lens in P containing a particular face in G, to accomplish step 5, could be done without altering the complexity T 3 (n i ) for faces meeting the geodesic arc in P. The other faces could be classified recursively by visiting faces which are adjacent to previously classified ones. Since we know the 1-lenses in P we can find their cones too. In this process we keep track of the number of terminal faces in each lens of P and finally we choose L as the 1- or 2-lens in P with zero terminal faces or the t scheme in P containing one terminal face. All this process could be completed in time T 5 (n i ) = O(n i ). Step 7 requires first to locate a chord in L or a loop in L and then determine where is the terminal (in the 3-lens or the 2-lens of L, or in a 1-lens contained in L), this process could be completed in O(n i ) time. Then we may need to empty a 3- or 2- lens, that would take additional time O(n 2 i ). After that we can bring out of L one edge and continue recursively. Since in this process we may require to make L singular, the number of steps is T 7 (n i ) = O(n 3 i ). In step 10 we need to identify if there is a 1- or 2-lens contained in L. This could be done by navigating all the geodesic arcs contained in L, we give to each geodesic arc a number and assign to each vertex the numbers of the geodesic arcs meeting at that vertex. A 1-lens is contained in L if and only if the pair of numbers assigned to a vertex are equal. A 2-lens is contained in L if and only if two vertices inside L have the same pair of numbers assigned. All this process could be completed in time proportional to the number of edges inside L which is O(n i ). Once we identify a 1- or 2-lens inside L we replace L by the new lens. We continue in this way until L becomes indecomposable, the whole process takes O(n 2 i ) time in the worst case. Finally we empty L by locating and ( Y) M transforming triangles incident to the boundary. We finish the process in T 10 (n i ) = O(n 2 i ) time. The total time we need to run the algorithm is T ( V (G 3 ) ) = V (G 3) n i=1 = O( V (G 3 ) 4 ) (T 3 (n i ) + T 4 (n i ) + T 5 (n i ) + T 7 (n i ) + T 10 (n i )) In this proof we made a direct implementation of the constructive approach in Theorem 5.1, the algorithm could be substantially improved by the application of specialized dynamic data structures and algorithms. In fact we conjecture that it could be improved to O( V (G 3 ) 3 ) worst time [7]. As a concluding remark observe that given the duality between the ( Y) M operations: {R0 M and R1 M }, {R2 M and R3 M }, {(Y ) M and ( Y) M } it is natural to define the dual operation (FP M ) 19

20 associated to FP M which can be applied to a subset of white faces distinguished as terminals. Then one may consider the problem of terminal ( Y) M reducing a medial graph with a set of distinguished black and white faces as terminals by using all of the eight operations on the medial graph. This is equivalent to the problem of Y reducing a plane graph where now we have subsets of vertices and faces as possible terminals. References [1] S. B. Akers Jr., The use of wye-delta transformations in network simplification, Operations Research, 8, (1960), [2] D. Archdeacon, C.J. Colbourn, I. Gitler and J.S. Provan, Four terminal reducibility and projective planar Y reducible graphs, J. Graph Theory 33, (2000), [3] Y. Colin de Verdiere, I. Gitler and D. Vertigan, Reseaux electriques planaires II, Comment. Math. Helvetici 71 (1996), [4] G. V. Epifanov, Reduction of a plane graph to an edge by a star-triangle transformation, Soviet Math. Doklady, 166, (1966), [5] T. A. Feo and J. S. Provan, Delta-wye transformations and the efficient reduction of two-terminal planar graphs, Oper. Res. 41 (1993), [6] I. Gitler, Delta-wye-delta transformations algorithms and applications, Ph.D. Thesis, Univ. Waterloo, [7] I. Gitler and F. Sagols, Algorithmic issues on planar terminal Y reducibility, preprint, [8] B. Grünbaum, Convex Polytopes, Interscience Publishers, London, [9] A.B. Lehman, J. Soc. Indust. Appl. Math. 11 (1963). [10] A. B. Lehman, Wye-delta transformations in probabilistic networks, J. SIAM, 11, (1962), [11] A. Schrijver, On the uniqueness of Kernels, J. Combin. Theory, Series B 55, (1992), [12] K. Truemper, On the delta-wye reduction for planar graphs, J. Graph Theory, 13, (1989),