Alliances and Bisection Width for Planar Graphs

Transcription

1 Alliances and Bisection Width for Planar Graphs Martin Olsen 1 and Morten Revsbæk 1 AU Herning Aarhs University, Denmark. martino@hih.a.dk MADAGO, Department of Compter Science Aarhs University, Denmark. mrevs@madalgo.a.dk Abstract. An alliance in a graph is a set of vertices (allies) sch that each vertex in the alliance has at least as many allies (conting the vertex itself) as non-allies in its neighborhood of the graph. We show that any planar graph with minimm degree at least can be split into two alliances in polynomial time. We base this on a proof of an pper bond of n on the bisection width for -connected planar graphs with an odd nmber of vertices. This improves a recently pblished n + 1 pper bond on the bisection width of planar graphs withot separating triangles and spports the folklore conjectre that a general pper bond of n exists for the bisection width of planar graphs. 1 Introdction An alliance is a set of vertices (allies) sch that any vertex in the alliance has at least as many allies (inclding the vertex itself) as non-allies in its neighborhood of the graph. The alliance is said to be strong if this holds even withot inclding the vertex itself among the allies. Alliances of vertices in graphs were introdced by Kristiansen et al. [11] to model among other things alliances of individals or nations bt appear many places in the literatre nder different names: Flake et al. [8] refer to a strong alliance as a commnity and base their work on the assmption that web pages related to each other form commnities in the web graph. Gerber and Kobler [10] look at what they refer to as the Satisfactory Graph Partition Problem where the objective is to partition a graph into two strong alliances. A partition of a graph into strong alliances can also be viewed as a so called Nash stable partition of an Additive Hedonic Game [13]. As mentioned above, alliances have been sed to model scenarios that might be planar of natre, so in this paper we focs on the problem of partitioning a planar graph into two alliances. In Section we show how to compte sch a partition in polynomial time for any planar graph with minimm degree at least. To prove this, we need an pper bond of n on the bisection width of -connected planar graphs with an odd nmber of vertices. We prove this pper bond in Section 3. Center for Massive Data Algorithmics, a center of the Danish National Research Fondation.

2 This tight pper bond is an improvement over the recently pblished [7] n + 1 pper bond for planar graphs withot separating triangles, and it spports the folklore conjectre [7], that a general pper bond of n exists for the bisection width of planar graphs. 1.1 Preliminaries Consider the connected graph G with vertex set V and edge set E where V = n and E = m. The degree d(v) of a vertex v in G is the nmber of edges incident to v in G. Similarly, for a sbset X V we define the degree d X (v) of a vertex v in the sbgraph of G indced by X {v} as d X (v) = { X : {v, } E}. We denote the minimm degree of the vertices in G as δ. A graph G is k-connected when at least k vertices are reqired to be removed in order to disconnect G. A cliqe is a flly connected graph and a maximal planar graph is a planar graph with the property that the addition of any new edge destroys planarity. An alliance in G is a non empty set A V sch that A : d A () + 1 d V A (). Throghot this paper when considering a planar graph, we will implicitly consider an embedding of the graph. A separating triangle in a planar graph is a triangle where both the interior and the exterior are non-empty. This definition can be tightened giving the notion of a strong alliance which is a non empty set A V sch that A : d A () d V A (). A partition of G is a collection of non-empty disjoint sbsets V 1... V l of V sch that l i=1 V i = V. For a partition of G into two sbsets V 1 and V we will denote the set of edges crossing this partition as e(v 1, V ) = {{, v} E : V 1 v V }. A bisection of G is a partition of G into V 1 and V sch that V 1 V 1 and the bisection width of G is defined as the minimm e(v 1, V ) over all bisections. 1. Related Work The problem of partitioning a graph into two strong alliances is NP-hard if we pt no restrictions on the graph []. There are however classes of graphs for which we can decide whether a partition into strong alliances exists and compte it in polynomial time. Examples of sch classes are graphs with maximm degree at most and graphs with girth at least 5 and minimm degree at least 3 [, 3]. For a general graph G, the comptational complexity of partitioning G into two alliances is an open problem []. Fricke et al. [9] show that any graph G contains an efficiently comptable alliance with no more than n vertices, while the problem of deciding whether an alliance with less than k members exists in G is NP-complete if k is part of the inpt. This even holds if G is planar [6]. Fan et al. [7] prove an pper bond of n + 1 for the bisection width for planar graphs withot a separating triangle and an pper bond on n for the bisection width for any triangle-free planar graph. The latter pper bond has sbseqently been strengthened by i et al. [1].

3 Alliances in Planar Graphs In this section we show that for planar graphs with minimm degree at least there exists a partition of the vertices into two alliances and that this partition can be compted in polynomial time. This is trivially also tre for planar graphs with minimm degree 1 (let one alliance consist of a single vertex with degree 1), while for planar graphs with minimm degree and 3 it is easy to find examples which show that not all sch graphs can be partitioned into alliances. See Figre 1. (a) (b) Fig. 1: Examples of planar graphs that can not be partitioned into alliances. In emma 1 we characterize a grop of general graph partitions that can be refined into an alliance partition sing a polynomial time algorithm. In the proof of Theorem 1 we then show how to constrct sch a partition for planar graphs with minimm degree in polynomial time. emma 1 is a precise formlation of the well known principle [, 9] that a partition into two sets of vertices forms a good starting point for obtaining a partition into alliances if the nmber of crossing edges is relatively small compared to the cardinality of the smallest set of vertices. emma 1. A graph G can be partitioned into two alliances if there exists a partition V 1, V of G sch that e(v 1, V ) min( V 1, V ) < δ. (1) The alliances can be compted in polynomial time if V 1 and V can be obtained in polynomial time. Proof. et V 1, V be a partition of G satisfying (1). We now rn the following simple algorithm: 1. et A 1 = V 1 and A = V.. If A 1 and A both are alliances or if one of them is empty we stop. Otherwise we go to step Assme that A 1 is not an alliance (otherwise we process A similarly). There mst be a A 1 with d A1 () + 1 < d A (). We now move from A 1 to A and go to step.

4 The nmber of crossing edges e(a 1, A ) decreases with or more every time step 3 is exected so the algorithm mst stop after no more than m steps. Assme that the algorithm stops becase A 1 is empty and let be the last vertex to leave A 1. We now consider the point in time where A 1 = {}: d V () = e(a 1, A ) e(v 1, V ) (min( V 1, V ) 1). We obtain a contradiction since (1) implies that the right hand side is less than δ. We conclde that the algorithm can not stop emptying A 1 or A. It has to stop with A 1 and A being alliances. Theorem 1. Any planar graph with δ can be partitioned into two alliances in polynomial time. Proof. We start by expanding the graph by adding edges ntil it is a maximal planar graph which can be done in polynomial time. We now consider two cases: The expanded graph has a separating triangle: A separating triangle has vertices both inside and otside of the triangle. et V 1 be the vertices on the side of the triangle containing the fewest vertices and let V = V \V 1. There can be no more than one vertex in V 1 having edges to all three vertices in the separating triangle so e(v 1, V ) V This ineqality also holds in the original graph so we can now se emma 1. The detection and processing of the separating triangle case is easily done in polynomial time. The expanded graph does not have a separating triangle: In this case the graph is -connected since all maximal planar graphs withot a separating triangle are -connected [5] and ths contains a hamiltonian cycle comptable in linear time [1]. Fan et al. [7] show how to efficiently compte a bisection V 1, V of V with e(v 1, V ) n + 1 for sch a graph. This makes it possible for s to apply emma 1 in the case where n is even bt for n odd an pper bond on n for the bisection width is needed to make ineqality (1) hold. In Section 3 we prove Theorem stating the existence of an efficiently comptable bisection V 1, V with e(v 1, V ) n for any -connected planar graph G(V, E) with an odd nmber of vertices. We now se emma 1 in the case where n is odd. As mentioned above, Fricke et al. [9] have shown that any graph contains an alliance with no more than n members. We can now improve this pper bond for planar graphs with δ : Corollary 1. Any planar graph with δ contains an alliance with no more than n members. 3 An Upper Bond for the Bisection Width We now show that a bisection V 1, V with e(v 1, V ) n can be compted in polynomial time for any -connected planar graph with an odd nmber of

5 vertices. Some of the techniqes sed are similar to the techniqes sed by Fan et al. [7] bt we also se other techniqes and the analysis is considerably more complicated compared to the analysis of Fan et al.. Since the bisection width never increases when removing edges from a graph, it is sfficient to only consider maximal -connected planar graphs with an odd nmber of vertices. emma. A maximal -connected planar graph with an odd nmber of vertices has a vertex with d() 5 sch that G is Hamiltonian. The vertex and the hamiltonian cycle of G can be fond in polynomial time. Proof. Consider a maximal -connected planar graph G with an odd nmber of vertices. There is at least one node in G with d() 5 since otherwise we wold have v V d(v) = m = (3n 6) n that cold only happen if n 5 which wold contradict -connectedness. The graph G is -connected so the graph G has a Hamiltonian cycle comptable in polynomial time as showed by Thomas and Y [1]. v W W t (a) (b) (c) (d) Fig. : Illstrations of a configration. Figre (a) shows the hamiltonian cycle with its k hamiltonian bisections (the dotted lines) and the cycle length of edge {v, t}. Figre (b) shows a single hamiltonian bisection where the vertices are colored according to which side of the bisection they belong to. Also, it shows and W for the configration. Figre (c) shows a compacted neighbor configration with and W. Figre (d) shows a heavy compacted neighbor configration where the dotted edges are the inner edges of the configration and the dashed edges are the oter edges of the configration. et G be a maximal -connected graph with an odd nmber of vertices and let be a vertex in G with d() 5 and C a Hamiltonian cycle in G. We will say that the tple (G,, C) represents a configration of G. For any sch configration, there are essentially k = n different ways to split C into two connected and eqally sized parts. From these parts, we constrct a hamiltonian bisection V 1, V of G by adding to the part where it has the most neighbors i.e. the part that minimizes e(v 1, V ) (ties are broken arbitrarily). Refer to Figre (a) and (b). In the following we let T (G,, C) denote the sm of e(v 1, V ) over the k possible hamiltonian bisections of (G,, C). We will sometimes omit

6 the argments if they are clear from the context. The cycle length of an edge {v, t} in G is the minimm distance between v and t in the graph indced by the cycle. The contribtion to T (G,, C) of an edge in G is precisely the cycle length of the edge. Refer to Figre (a). We let denote the length of the longest path along C starting and ending at a neighbor from bt visiting no other neighbors of and let W denote the length of the second longest sch path. Refer to Figre (b). We refer to the configration (G,, C) as a compacted neighbor configration if the neighbors of can be divided into two sbsets N 1 and N of size d() and respectively sch that each sbset occpies a connected sbpath of d() the hamiltonian cycle C. Refer to Figre (c). The inner edges are the edges on the same side of C as. The inner edges that are not incident to are natrally groped into (at most) two grops in a compacted neighbor configration. A compacted neighbor configration is called heavy if the edges from both these grops have cycle lengths, 3,,..., k, k 1, k, k 3,... (for both grops we start the seqence from the left) and if the set of oter edges has two edges of length, two edges of length 3,..., two edges of length k 1 and one edge of length k. Refer to Figre (d). In what follows, we will show that T (G,, C) < k(n+1) for any configration (G,, C) of a maximal -connected planar graph with an odd nmber of vertices. Since T (G,, C) is the sm of bisection sizes for the k hamiltonian bisections this implies that there exists at least one hamiltonian bisection V 1, V sch that e(v 1, V ) n which then gives s the pper bond on the bisection width. To prove T (G,, C) < k(n + 1) we will first show that the heavy compacted neighbor configrations can be considered as a set of worst case configrations sch that for any configration (G,, C) there exists a heavy compacted neighbor configration (G,, C ) where T (G,, C) T (G,, C ). We then exploit that the heavy compacted neighbor configrations are reasonably simple sch that T (G,, C ) < k(n + 1) can be shown for this set of configrations. Ŵ ˆ W Fig. 3: In the configration (G,, C) to the left, the hamiltonian bisections where edges incident to contribte with d()/ are shown with dotted lines. Similarly, in the configration (Ĝ,, C) to the right, the hamitonian bisections where edges incident to contribte with d( )/ are shown with dotted lines.

7 ˆ ˆ (a) (b) Fig. : In (a) we show those hamiltonian bisections which flly contain the vertices on the cycle path corresponding to in (G,, C) (to the left) and in (Ĝ,, C) (to the right). In (b) we show those hamiltonian bisections which does not flly contain the vertices in (G,, C) (to the left) and in (Ĝ,, C) (to the right). emma 3. If (G,, C) is a configration then it is possible to constrct a heavy compacted neighbor configration (G,, C) where G and G have the same nmber of vertices and d() = d( ) sch that T (G,, C) T (G,, C). Proof. et (G,, C) represent an arbitrary configration. We now remove those edges in G that are not on C and not incident to. We then replace (and the edges incident to ) with a vertex with d() = d( ) sch that the reslting configration (Ĝ,, C) is a compacted neighbor configration. Finally, we pt in edges to create the graph G sch that (G,, C) is a heavy compacted neighbor configration. Below, we first arge that the contribtion to T of edges incident to in G is not higher than the contribtion to T of edges incident to in G. Secondly, we arge that the contribtion to T of edges in G is not higher than the contribtion to T of edges in G. Edges incident to : We separate or analysis into a case analysis based on the vale of in G. The vales of and W in Ĝ are denoted by ˆ and Ŵ respectively. Case 1: k: We consider the following sbcases: If + d() k we bild the compacted neighbor configration (Ĝ,, C) sch that ˆ Ŵ is minimized (0 or 1). Refer to Figre 3. The contribtion to T of edges incident to is k which is the d() maximm obtainable vale since always chooses to join the partition which contribtes the least to T. Refer to Figre 3. The condition + d() k makes it possible for s to obtain the k contribtion d() to T from edges incident to and at the same time obtain ˆ and Ŵ W that is important when we consider the contribtion from the other edges. If + d() > k we bild the compacted neighbor configration (Ĝ,, C) sch that = ˆ and sch that the nodes forming the long paths along C with no neighbors of of respectively are the same. Refer to Figre. For each of the k hamiltonian bisections in (Ĝ,, C)

8 we now show that the nmber of crossing edges incident to has not decreased compared to the corresponding (same partition of C) hamiltonian bisection in (G,, C). - We first consider a bisection V 1, V where the vertices on the path along C of length = ˆ is flly contained within either V 1 or V say V 1. In this case, mst choose to join V. The nmber of neighbors of in V 1 is at least as high as the nmber of neighbors of in V 1 in G so the nmber of crossing edges for sch a bisection has not decreased. Refer to Figre (a). - We now consider a bisection where the vertices on the path along C of length = ˆ are not flly contained within either side of the bisection. When has chosen a side of the bisection has only crossing edges to members of either N 1 or N (the two grops of neighbors of ). If has d( ) crossing edges the case is clear. Otherwise, the nmber crossing edges has not dropped since every node on the other side of the ct and not on the long path is a neighbor to. Refer to Figre (b). Case : > k: We bild the compacted neighbor configration (Ĝ,, C) with = ˆ. Consider a bisection V 1, V of (Ĝ,, C). When chooses side of the bisection can not have crossing edges to both N 1 and N. If there are no crossing edges the same wold be the case for the corresponding bisection of the original configration. Refer to Figre 5(a). If there are crossing edges then the nmber of neighbors on the other side can not have decreased. Refer to Figre 5(b) Edges not incident to : Since C is in both G and G the edges on C obviosly contribte with the same to T. We now consider the edges in G not incident to and not on C and the edges of G not incident to and not on C. Fan et al. [7] show how to eliminate any triangle of sch edges and obtain a new set of edges with higher cycle lengths by replacing some of the edges and Fan et al. also arge that repeated elimination of triangles will prodce a heavy configration we refer to [7] for more details. The fact that ˆ and Ŵ W makes it possible to se this techniqe and obtain a one-to-one correspondence between the two sets of edges considered sch that any edge in the G-set is matched with an edge in the G -set with the same cycle length or a bigger cycle length. The contribtion to T of these edges can conseqently not decrease dring the transformation. emma. et (G,, C) be a heavy compacted neighbor configration with d() even. The contribtion to T (G,, C) of the edges incident to is d() + W d() d() Proof. We grop the edges incident to into pairs sch that a pair of edges cts C into two pieces with the same nmber of neighbors of. For a given.

9 ˆ ˆ (a) (b) Fig. 5: In (a) we illstrate the case where there are no crossing edges for a hamiltonian bisection in G and the corresponding bisection of Ĝ. In (b) we show the bisections where there are crossing edges in which case the nmber of crossing edges can not have decreased in Ĝ. hamiltonian bisection the contribtion to T (G,, C) of a pair is 1 if the endpoints of the edges are separated and 0 otherwise. There are W + d() 1 bisections that separate each pair so it is now easy to compte the contribtion to T (G,, C) of the edges incident to : d() d() (W + 1). emma 5. If (G,, C) is a heavy compacted neighbor configration then we have the following: T (G,, C) < k(n + 1). Proof. We divide the proof into three cases. Assme that k 1 and that d() is even: We compte T in the following way: ( ) k 1 k 1 d() 3 ( d() T = k+ k + i + k + i (W + i) + + W d() d() ) i= i= The first term is the sm of cycle lengths from the edges on the cycle, the second term is the sm of cycle lengths for the oter edges, the third term is the sm of cycle lengths for the inner edges not incident to, and the forth term is the contribtion from edges incident to given by emma. We now se ( ) k 1 i= i = (k 1)k 1 and n = k + 1: ( d() T k(n + 1) = + W d() i=1 d() ) d() 3 (W + i). We now work on a part of this sm mltiplied by in order to exclsively have integers in the comptation: ( d() + W d() d() ) d() 3 (W + i) i=1 i=1.

10 = (d() )d() + W d() (d() 3)W (d() 3)(d() ) = d() +8d() 1+1W W d() = (d() 6)(d() )+1W W d() = d() + 8d() 1 + 1W W d() = (d() 6)(d() + W ), and finally we get (d() 6)(d() + W ) T k(n + 1) = < 0, () where we have sed that the degree of is at least 6. Now assme that k and that d() is even: In this case we get T = W i + i + d() i= ( + 1) = 1 + implying i= W (W + 1) + W d() 1 + d() d() + W d() + k + k d() + k + k T k(k+) = (+1) 8+W (W +1)+d() +W d() d()+8k+k 8 k(k+) = ( + 1) + W (W + 1) + d()(d() + W ) k 16. We now se W + + d() = k: T k(k+) = (+1)+W (W +1)+(k+ W )(k+w ) k 16 We now se W in (3): implying = 3 + W k + W + k 16. (3) T k(k + ) + ( k) + k 16 = (( 1)( k + ) ). T k(n + 1) ( 1)( k + ) < 0 for {1,,..., k }. Now assme that d() is odd: We remove the edge of from the grop with edges that is closest to the path along the cycle corresponding to W. It d() is not hard to see that the contribtion to T of the edges of is nchanged after the removal of this edge. For d() > 5 there is conseqently a heavy compacted neighbor configration considered above with a higher vale of T compared to the original graph. If d() = 5 we can se () with d() = and W replaced by W + 1 if we sbtract W + 1 (when the edge of is removed as described above we insert an edge with cycle length W + 1 and obtain a heavy compacted neighbor configration with d() = ): ( 6)( + (W + 1)) T k(n + 1) = (W + 1) = 3 < 0.

11 Theorem. A bisection V 1, V exists with e(v 1, V ) n for any -connected planar graph G(V, E) with an odd nmber of vertices and sch a bisection can be obtained in polynomial time. Proof. et G(V, E) be a -connected planar graph with an odd nmber of vertices. As noted earlier, we can assme that G is a maximal planar graph withot loss of generality. emma assres that we can efficiently obtain a configration (G,, c). We now examine all the k hamiltonian bisections of the configration. By sing emma 3 and emma 5 we know that at least one of the hamiltonian bisections satisfies e(v 1, V ) n. References 1. Takao Asano, Shnji Kikchi, and Nobji Saito. A linear algorithm for finding hamiltonian cycles in -connected maximal planar graphs. Discrete Applied Mathematics, 7(1):1 15, Cristina Bazgan, Zsolt Tza, and Daniel Vanderpooten. The satisfactory partition problem. Discrete Appl. Math., 15:136 15, May Cristina Bazgan, Zsolt Tza, and Daniel Vanderpooten. Efficient algorithms for decomposing graphs nder degree constraints. Discrete Appl. Math., 155(8): , Cristina Bazgan, Zsolt Tza, and Daniel Vanderpooten. Satisfactory graph partition, variants, and generalizations. Eropean Jornal of Operational Research, 06():71 80, Chiyan Chen. Any maximal planar graph with only one separating triangle is hamiltonian. J. Comb. Optim., 7(1):79 86, Rosa I. Enciso. Alliances in graphs: parameterized algorithms and on partitioning series-parallel graphs. PhD thesis, University of Central Florida, Gengha Fan, Baogang X, Xingxing Y, and Chixiang Zho. Upper bonds on minimm balanced bipartitions. Discrete Mathematics, 31(6): , Gary Flake, Steve awrence, and C. ee Giles. Efficient identification of web commnities. In Proc. 6th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages ACM Press, G. H. Fricke,. M. awson, T. W. Haynes, S. M. Hedetniemi, and S. T. Hedetniemi. A note on defensive alliances in graphs. Blletin ICA, 38:37 1, Michael U. Gerber and Daniel Kobler. Classes of graphs that can be partitioned to satisfy all their vertices. Astralasian Jornal of Combinatorics, 9:01 1, P. Kristiansen, S. M. Hedetniemi, and S. T. Hedetniemi. Alliances in graphs. Jornal of Combinatorial Mathematics and Combinatorial Compting, 8: , Haiyan i, Yanting iang, Mho i, and Baogang X. On minimm balanced bipartitions of triangle-free graphs. Jornal of Combinatorial Optimization, pages 1 10, Shao Chin Sng and Dinko Dimitrov. Comptational complexity in additive hedonic games. Eropean Jornal of Operational Research, 03: , R. Thomas and X. X. Y. -connected projective-planar graphs are hamiltonian. Jornal of Combinatorial Theory, Series B, 6(1):11 13, 199.