Abstract
We obtain a complete classification of graphs H for which the class of (triangle,H)-free graphs is well-quasi-ordered by the induced subgraph relation and an almost complete classification of graphs H for which the class of (triangle,H)-free graphs has bounded clique-width. In particular, we show that for these graph classes, well-quasi-orderability implies boundedness of clique-width. To obtain our results, we further refine a known method based on canonical decomposition. This leads to a new decomposition technique that is applicable to both notions, well-quasi-orderability and clique-width.
Highlights
In a previous paper [15] we showed that certain graph operations and graph constructions work well for bounded clique-width and well-quasi-orderability
Instead of using ad hoc arguments, we emphasize that our underlying research goal is to develop methodology that could be applied more widely to other classes and which increases our understanding of the structure of special graph classes
We prove our results on both well-quasi-orderability and boundedness of clique-width by a further generalization of the canonical decomposition method introduced by Fouquet, Giakoumakis and Vanherpe [24] for bipartite graphs
Summary
The vertex set decomposition of clique-width is defined via a graph construction based on vertex labellings. The algorithmic importance of having bounded clique-width follows from the existence of several meta-theorems [10,21,31, 41] which, when combined with an approximation result [39], ensure that many well-known NP-hard graph problems, such as Graph Colouring and Hamilton Cycle, become polynomial-time solvable for every graph class of bounded clique-width. There is a need to verify boundedness of clique-width of special graph classes, in particular when undertaking a systematic classification into the computational complexity of graph problems under input restrictions; see, for example, [25] for the importance of clique-width for the Graph Colouring problem
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