Let be an symmetric matrix over , we will call such a matrix a pattern. An -partition of a graph is a partition of such that, for every , every vertex in is adjacent to every vertex in if , every vertex in is non-adjacent to every vertex in if , and there are no restrictions between vertices of and vertices of if . Notice that in the previous definition we might have , in which case we will have that is a clique if , and is an independent set if . As it is common in Graph Theory, we do not ask every part in our partitions to be non-empty, so we will usually forbid the diagonal entries of our patterns to be ; if for some , then we would have a part without restrictions, so we might place all the vertices in to obtain a trivial -partition. A graph that admits an -partition is called -partitionable.
Many classical problems in Graph Theory can be stated as -partition problems, e.g., the matrix partition problem corresponding to a pattern with all diagonal entries equal to and all off-diagonal entries equal to is just the usual -colouring problem; for every pattern without ’s, the -partition problem corresponds to an -homomorphism problem, where is the graph whose adjacency matris is obtained from by replacing each by a . The interested reader may refer to hellEJC35 for a wonderful survey on the subject. Clearly, having an -partition is a hereditary property, and thus, for a fixed pattern , the class of -partitionable graphs can be characterized in terms of a family of forbidden induced subgraphs. A graph is a minimal obstruction for the -partition problem if it is not -partitionable, but every proper induced subgraph is. The set of all minimal obstructions for an -partition problem is, besides a family of forbidden induced subgraphs characterizing -partitionable digraphs, also a set of no-certificates for an algorithm that verifies if a graph is -partitionable. Recall that a certifying algorithm is an algorithm for a decision problem which provides a yes-certificate for every yes-instance of the problem, and a no-certificate for every no-instance of the problem. If the validity of these certificates can be verified faster than the time it takes to actually solve the problem, then they provide a valuable tool to check the correctness of an implementation of the algorithm.
In hellEJC35 , two main problems related to matrix partitions are discussed. The Characterization Problem asks which patterns have the property that the number of minimal obstructions to -partition is finite. The Complexity Problem asks which patterns have the property that the -partition problem can be solved by a polynomial time algorithm. Since both problems are very hard in the general case, it has been a common practice to restrict them to well behaved families of graphs. Recall that a graph is perfect if for every induced subgraph, its chromatic number coincides with the size of a maximum clique. As perfect graphs have a unique minimal obstruction for the -colouring problem (the complete graph on vertices), it is usual to consider a family of perfect graphs, like cographs or chordal graphs. It is known that the answer for the characterization problem is always positive for cographs damaschkeJGT14 ; federGTP2006 and split graphs federDAM166 . A graph is chordal if every cycle in has a chord, or equivalently, if every induced cycle in has length . The class of chordal graphs is one of the best understood graph families where these two problems are still open. Many papers deal with matrix partition problems for chordal graphs, e.g., in hellDAM141 , a forbidden subgraph characterization and a polynomial time recognition algorithm is given for chordal graphs admitting a partition into independent sets and cliques, federTCS349 considers the list version of the problem on chordal graphs, polarity of chordal graphs is studied in ekimDAM156 , in hellDM338 the -partition problem is studied on chordal graphs for a special family of patterns called joining matrices. In particular, this work might be thought as a complement of federDM313 , where both the characterization problem and the complexity problem is studied for small matrices ( with ) on chordal graphs. In particular, they show that if is a matrix of size , then has finitely many chordal minimal obstructions, except for the following two matrices, which have inifinitely many chordal minimal obstructions.
In federDM313 , an infinite family of chordal minimal obstructions is exhibited for these patterns (the same family works for both patterns). Despite the fact that the complete family of minimal obstructions is obtained for other patterns, no efforts are made to obtain the complete list for patterns and . The main objective of the present work is to provide this missing list for . As we will see, among the minimal obstructions there is a single infinite family, all the other obstructions occur sporadically.
Let be the family of chordal graphs containing for , and the members of the family , depicted in Figure 1. Notice that the graphs in
consist of an odd path of length at least, together with an additional vertex adjacent to every vertex of the path, except for the first and last.
The main result of this work is the following theorem.
If is a chordal graph, then admits an -partition if and only if it is -free.
For basic notions we refer the reader to bondy2008 . Given a set of graphs, we say that a graph is -free if it does not contain any member of as an induced subgraph. When we will abuse notation and say that is -free.
2 Preliminary results
In this section we will obtain some basic technical results, necessary for the proof of our main theorem.
Let be a chordal graph. If contains as a subgraph, then it contains or as an induced subgraph.
Consider a copy of as a subraph of with the configuration shown in Figure 2.
If there are no further edges, this is, if the dotted edges are missing in the figure, then this is is an induced copy of . If any of the edges or is present, then we have as an induced subgraph. If is present, then we also have the cycle and, since is chordal, either or must be an edge of , and we are in the previous case. The cases when or are present are analogous.
Our following lemma has a similar flavour.
Let be a chordal graph. If contains as a subgraph, then it contains or as an induced subgraph.
Consider a copy of as a subraph of with the configuration shown in Figure 3.
If there are no further edges, this is, if the dotted edges are missing in the figure, then this is is an induced copy of . If any of the edges , or is present, then we have as an induced subgraph. If the edge is present, the cycle , is contained in , and it follows from the chordality of that either or is an edge, which takes us to the previous case. An analogous case happens if is an edge of . Finally, if is present, then is a cycle in , and thus or must be an edge of . Again, this falls in one of the previous cases.
Although again, similar in nature, our following lemma deals with a graph not in . For consistency, we will denote by .
Let be a chordal graph. If contains as a subgraph, then it contains , or as an induced subgraph.
Let and be the two triangles in . If there are less than three edges between and , then, since is chordal, they cannot form a matching (otherwise there would be a chordless in ). Thus, contains an induced copy of .
So, suppose that there are at least three edges between and . If three edges are incident in the same vertex of , then the vertices of together with induce an (left side of Figure 4). Else, at least two of the edges from to form a matching, and hence a with one edge from each triangle. Again, the chordality of implies that at least one of the diagonals of the exists, and thus, the result follows from Lemma 3 (right side in Figure 4).
The following result characterizes chordal graphs which are bipartizable by removing a single vertex, in terms of forbidden subgraphs.
If is a chordal graph, then there exists a vertex such that is bipartite if and only if contains neither , , nor as a subgraph (not necessarily induced).
The result is clear if contains a copy of , or . For the converse we will proceed by contradiction.
Aiming for a contradiction, suppose that contains neither , nor as subgraphs, and for every vertex of , we have that is not bipartite. It follows that contains at least two different triangles and . Since does not contain as an induced subgraph, then and must share one, or two vertices.
Suppose first that and share a single vertex . Since is not bipartite, then there is another triangle in which does not contain the vertex , but shares vertices with both and . If shares exactly on vertex with , and exactly on vertex with , then its third vertex should be a new vertex, neither in nor in , which would give us a copy of contained in , which cannot happen (see leftmost graph in Figure 5). If shares two vertices with and a single vertex with , then would contain a copy of as a subgraph (center graph in Figure 5). Thus, this case is also impossible.
If and share two vertices, say and , then, since is not bipartite, there exists a triangle in not using the vertex and such that it shares two vertices with and two vertices with (otherwise we would be in the previous case). But for this to happen, there should be an edge joining the vertices in and different from and (rightmost graph in Figure 5), which creates a copy of , a contradiction.
Since a contradiction is reached in each case, we conclude that there is a vertex in such that is bipartite.
Notice that Lemma 5 can be easily adapted to be stated in terms of forbidden induced subgraphs, as our next result shows. Besides the graphs in family , we need two additional graphs, and , depicted in Figure 6.
If is a chordal graph, then there exists a vertex such that is bipartite if and only if is -free.
We will only prove the non-trivial implication. Suppose that is a chordal -free graph. As is -free, it follows from Lemma 2, that does not contain as a subgraph. Since is complete, having it as an induced subgraph is equivalent to having it as a subgraph.
In the proof of Lemma 4, in the case where there are at least three edges between and , we concluded that contains an induced copy of either or . It is not hard to verify that in the case where there are at most two edges, must contain an induced copy of either or .
Thus, if is -free, then it contains neither nor as a subgraph. The desired result follows from Lemma 5. ∎
Clearly, if is not an induced subgraph of , then neither and are. Thus, a direct application of Lemma 6 produces the next result.
If is a chordal graph, then is -free if and only if is )-free and there exists a vertex in such that is bipartite.
3 Main result
For a chordal graph we define the set as
then every element of must belong to every triangle in , and hence, if is not bipartite, then .
Notice that if a graph admits an -partition, then any graph obtained from by adding isolated vertices also admits an -partition; it suffices to place all the isolated vertices in , the independent set without further restrictions. Also, if is a chordal, non-bipartite, -free graph, then at least one component of contains a triangle, and being -free, every other component of must be an isolated vertex. Thus, in order to prove our main result, we may only consider connected graphs.
If is a chordal graph, then admits an -partition if and only if is -free and there exists a vertex such that is bipartite.
Let be a graph with an -partition (recall that and are completely adjacent). If is bipartite, then the conditions of the theorem clearly hold. Else, is non-empty for . If , then would contain a chordless -cycle, contradicting the chordality of . Thus, we will assume without loss of generality that . Hence, is a bipartite graph.
For the remaining condition, it suffices to verify that none of , nor any graph in admits an -partition. These verifications are simple yet tedious, so we will omit them. This concludes the proof for the necessity.
For the suficiency, if is bipartite, then it clearly admits an -partition. Else, as we discussed at the beginning of this section, we can assume that is connected. Since we will consider three cases, one for each possible cardinality of . We are looking for an -partition as described in the Introduction; we will describe such a partition by by colouring the vertices of with colours , which correspond to the parts of the partition.
Case 1: If , then, as each of and must be in every triangle of , then there is only one triangle in , namely the one induced by .
Since is chordal, is acyclic for every , as otherwise there would be a triangle not using . Let be the connected component of where . Then, is a tree for , and since is -free, then for at least one . Assume without loss of generality that .
If we consider as a tree rooted at , then the fact that is -free implies that and cannot both have height greater than . Suppose without loss of generality that has height less than . If has height greater than , then it must be , otherwise would contain as an induced subgraph. See the leftmost graph in Figure 7 for a depiction of the current configuration.
Thus, we colour with colour , with colour and with colour . The neighbours of in are coloured with , the neighbours of in are coloured , and the vertices at depth in are coloured . It follows from the structure of in this case that this colouring is an -partition.
Case 2: Suppose that . If there were exactly one triangle in , then all of its vertices would be in , and we would be back to Case 1. So, there are at least two different triangles in . Also, recall that and are in every triangle of . Let be a vertex such that is a triangle in . Let be the connected component of the subgraph of obtained by deleting and all the vertices that form a triangle together with and , which contains . The chordality of and the definition of imply that is a tree, and thus we consider to be a tree rooted at . The subgraph is analogously defined. Also analogously, the graph is a forest. Let be the connected component containing in , and consider it as a tree rooted at . Since there is a triangle in other than , and is -free, then (and for any vertex forming a triangle together with and ) has height at most .
Case 2.1: As a first subcase, suppose that has height . Since is -free, then or must have height . Assume without loss of generality that has height . Since is -free, then has height at most . See the center picture in Figure 7.
Thus, colour with colour , with colour , every vertex forming a triangle together with and with colour , and every vertex at depth in for every such with colour . Also, colour every vertex in at depth with colour , and every at depth (if any) with colour . By the previous analysis of the structure in this case, we have that this colouring induces an -partition of .
Case 2.2: For every vertex forming a triangle with and , the rooted tree has height . Since is -free, we assume without loss of generality that has height , and thus, might have height (see the rightmost graph in Figure 7).
Hence, we colour with colour , with colour , every vertex forming a triangle with and with colour , every neighbour of other than with colour , the vertices at depth in with colour and the vertices at depth in with colour .
Case 3: Finally, suppose that . As in the previous case, there must be at least two different triangles in . We will first show that the eccentricity of in is . To reach a contradiction, suppose that is a vertex at distance from , and let be a path in realizing this distance. Since and are not adjacent to , then they are not in any triangle of , and thus, they belong to no cycle of . Since , then there must be a triangle in (which as mentioned, uses neither nor ). Additionally, the edge is not adjacent to , since otherwise, as uses , there would be a cycle in using or . But now, the vertices of together with and induce a copy of , a contradiction. Thus, the eccentricity of is at most .
Let be the set of all vertices at distance from , with . From the discussion in the previous paragraph, we have that . Notice that is independent, as otherwise an edge between vertices of would necessarily belong to a cycle, and hence to a triangle of , which in turn would imply that its vertices are adjacent to , contradicting the definition of . So, the neighbourhood of every vertex in is contained in . Moreover, since vertices of do not belong to any cycle of , then they have degree . Again, since is in every triangle of , then the induced subgraph is a forest, and hence each of its connected components is uniquely -coloureable.
Suppose that there are vertices and vertices such that , and are edges in . See Figure 8 for the configuration of this case.
As is not in , there is triangle in . If the two vertices in other than are different from , then at least one of them is non-adjacent to (recall that is acyclic), and hence, this vertex together with , and induce a copy of . Thus, one of the vertices of is ; let us call the other vertex (see Figure 8).
Repeating the above procedure for , there must exist another triangle not containing , using , and another vertex , different from . Recall that since is acyclic, then and . Thus, induces a copy of , a contradiction.
Hence, if vertices have neighbours in , then they cannot be adjacent. Moreover, if and have neighbours in , as is -free, then the distance between and in is even. Thus, in every connected component in , the vertices with neighbours in are in the same part of the bipartition.
It follows from the previous description of the structure of that the following colouring induces an -partition of . Colour with colour and every element of with colour . For every connected component of , if it has any adjacencies with , (properly) -colour it in such a way that each vertex with neighbours in receives colour ; else, -colour it in any way.
Since the cases are exhaustive, the desired result follows. ∎
4 Conclusions and further work
We have provided a complete list of minimal obstructions for one of the only two matrices which have inifinitely many chordal minimal obstructions. This nearly completes our knowledge on the matrix partition problem for all matrices on chordal graphs. In federTCS349 , patterns with NP-complete -partition problems for chordal graphs are constructed, but they are rather large (close to thirty rows and columns). Hopefully, a complete understanding of the matrix partition problem for small patterns will lead us to find the smallest pattern having an NP-complete -partition problem.
As the reader may notice, the exact list of chordal minimal obstructions is still missing for . This list would complete the analysis of all patterns. Fortunately, a complete list has been already obtained as part of the first author’s Ph.D. thesis, and a follow up article including this result is in preparation. An interesting situation arises for , there are two different infinite families of chordal minimal obstructions for the -partition problem.
We would like to thank Sebastián González Hermosillo de la Maza for pointing out Lemma 5 to us.
- (1) A. Bondy and U.S.R. Murty. “Graph Theory”. Springer-Verlag, 2008.
- (2) P. Damaschke, Induced subgraphs and well-quasi-ordering, Journal of Graph Theory 14 (1990) 427–435.
- (3) T. Ekim, P. Hell, J. Stacho and D. de Werra, Polarity of chordal graphs, Discrete Applied Mathematics 156(13) (2008) 2469–2479.
- (4) T. Feder, P. Hell and W. Hochstättler, Generalized colourings of cographs, in Graph Theory in Paris, Birkhäuser Verlag 2006, pp. 149–167.
- (5) T. Feder, P. Hell, S. Klein, L. T. Nogueira and F. Protti, List matrix partitions of chordal graphs, Theoretical Computer Science 349 (2005) 52–66.
- (6) T. Feder, P. Hell and S. N. Rizi, Obstructions to partitions of chordal graphs, Discrete Mathematics 313 (2013) 1861–1871.
- (7) T. Feder, P. Hell and O. Shklarsky, Matrix partitions of split graphs, Discrete Applied Mathematics 166 (2014) 91–96.
- (8) P. Hell, Graph partitions with prescribed patterns, European Journal of Combinatorics 35 (2014) 335–353.
- (9) P. Hell, S. Klein, L. T. Nogueira and F. Protti, Partitioning chordal graphs into independent sets and cliques, Discrete Applied Mathematics 141(1–3) (2004) 185–194.
- (10) P. Hell and P.-L. Yen, Join colourings of chordal graphs, Discrete Mathematics 338(12) (2015) 2453–2461.