Infinitely many minimal classes of graphs of unbounded clique-width

Abstract

The celebrated theorem of Robertson and Seymour states that in the family of minor-closed graph classes, there is a unique minimal class of graphs of unbounded tree-width, namely, the class of planar graphs. In the case of tree-width, the restriction to minor-closed classes is justified by the fact that the tree-width of a graph is never smaller than the tree-width of any of its minors. This, however, is not the case with respect to clique-width, as the clique-width of a graph can be (much) smaller than the clique-width of its minor. On the other hand, the clique-width of a graph is never smaller than the clique-width of any of its induced subgraphs, which allows us to be restricted to hereditary classes (that is, classes closed under taking induced subgraphs), when we study clique-width. Up to date, only finitely many minimal hereditary classes of graphs of unbounded clique-width have been discovered in the literature. In the present paper, we prove that the family of such classes is infinite. Moreover, we show that the same is true with respect to linear clique-width.the clique-width of G. On the other hand, the clique-width of G is never smaller than the clique-width of any of its induced subgraphs [5]. This allows us to be restricted to hereditary classes, that is, those that are closed under taking induced subgraphs.One of the most remarkable outcomes of the graph minor project of Robertson and Seymour is the proof of Wagner's conjecture stating that the minor relation is a well-quasiorder [13]. This implies, in particular, that in the world of minor-closed graph classes there exist minimal classes of unbounded tree-width and the number of such classes is finite. In fact, there is just one such class (the planar graphs), which was shown even before the proof of Wagner's conjecture [12].In the world of hereditary classes the situation is more complicated, because the induced subgraph relation is not a well-quasi-order. It contains infinite antichains, and hence, there may exist infinite strictly decreasing sequences of graph classes with no minimal one. In other words, even the existence of minimal hereditary classes of unbounded clique-width is not an obvious fact. This fact was recently confirmed in [8]. However, whether the number of such classes is finite or infinite remained an open question. In the present paper, we settle this question by showing that the family of minimal hereditary classes of unbounded clique-width is infinite. Moreover, we prove that the same is true with respect to linear clique-width.The organisation of the paper is as follows. In the next section, we introduce basic notation and terminology. In Section 3, we describe a family of graph classes of unbounded clique-width and prove that infinitely many of them are minimal with respect to this property. In Section 4, we identify more classes of unbounded clique-width. Finally, Section 5 concludes the paper with a number of open problems.

Clique-width for hereditary graph classes

“…First, the classes of split permutation graphs (and the analogous bipartite class of bichain graphs) [5], unit interval graphs [136] and bipartite permutation graphs [136] are even minimal hereditary graph classes of unbounded clique-width. Collins, Foniok, Korpelainen, Lozin and Zamaraev [48] proved that the number of minimal hereditary graphs of unbounded clique-width is infinite. Second, for classes, such as split graphs, bipartite graphs, co-bipartite graphs and (K 1,3 , 2K 2 )-free graphs, unboundedness of clique-width also follows from the fact that these classes are superfactorial [18] and an application of the following result.…”

Uncountably many minimal hereditary classes of graphs of unbounded clique-width

“…Lemma 3.1 (Collins, Foniok, Korpelainen, Lozin and Zamaraev [3]). If α is an infinite binary word containing infinitely many 1s then the graph class G α has unbounded clique-width.…”

“…For this we will use the rank-width parameter described in Section 2.2. From [3] we have a toolkit of graph operations which we extend to show that the graph class G α contains a graph with a vertex-minor H γ 1,1 (q, q) for some q where γ is an infinite word from the alphabet {2, 3}. If we can make q as large as we like then combining Lemma 2.1 with Lemma 3.7 gives us the result that G α has unbounded rank-width and therefore unbounded clique-width.…”

Between clique-width and linear clique-width of bipartite graphs

“…Recently, many classes of graphs have been shown to be of bounded clique-width, and for many others, the clique-width was shown to be unbounded, see e.g. [7,8,10,22,24]. Most of these studies concern hereditary classes, i.e.…”

“…In a similar way, in the study of clique-width of particular importance are minimal classes of graphs of unbounded clique-width. The first two hereditary classes of this type have been identified in [20] and only recently it was shown in [10] that the number of such classes is infinite. What is interesting is that all the classes found in [10] and in [20] are also minimal hereditary classes of unbounded linear clique-width.…”