Graph classes with and without powers of bounded clique-width

Abstract

We initiate the study of graph classes of power-bounded clique-width, that is, graph classes for which there exist integers $k$ and $\ell$ such that the $k$-th powers of the graphs are of clique-width at most $\ell$. We give sufficient and necessary conditions for this property. As our main results, we characterize graph classes of power-bounded clique-width within classes defined by either one forbidden induced subgraph, or by two connected forbidden induced subgraphs. We also show that for every positive integer $k$, there exists a graph class such that the $k$-th powers of graphs in the class form a class of bounded clique-width, while this is not the case for any smaller power.Comment: 23 pages, 4 figure

Classifying the clique-width of H-free bipartite graphs

“…In particular, many papers have determined the clique-width of graph classes characterized by one or more forbidden induced subgraphs [1,2,[5][6][7][8][9][10][11][12]15,20,[23][24][25][26].…”

Bounding the clique-width of H-free split graphs

“…In particular we refer to [19] for details on how new results can be combined with known results to give a classification for all but 13 open cases (up to an equivalence relation). Similar studies have been performed for variants of clique-width, such as linear clique-width [26] and power-bounded clique-width [2]. Moreover, the (un)boundedness of the cliquewidth of a graph class seems to be related to the computational complexity of the Graph Isomorphism problem, which has in particular been investigated for graph classes defined by two forbidden induced subgraphs [28,33].…”

“…Furthermore, we make use of other results that do not give explicit bounds. 2 Note that the boundedness in Theorems 9 and 11 also follows from Lemma 5 and Theorem 2, combined with the fact that every split graph is chordal. However, the proof of Theorem 2 in [3] relies on results from this paper.…”

“…A graph class is said to be of bounded clique-width if there is a constant c such that the clique-width of every graph in the class is at most c. Much research has been done identifying whether or not various classes have bounded clique-width [1][2][3][5][6][7][8][9][10][11][15][16][17][18][19]23,[29][30][31][32]. For instance, the Information System on Graph Classes and their Inclusions [20] maintains a record of graph classes for which this is known.…”

Bounding clique-width via perfect graphs

“…By applying our technique, we are able to present three new classes of (H 1 , H 2 )-free graphs of bounded cliquewidth. 2 By modifying walls via graph operations that preserve unboundedness of clique-width, we are also able to present a new class of (H 1 , H 2 )-free graphs of unbounded clique-width. Combining our results leads to the following theorem (see also Fig.…”