Abstract:
Clique-width is a graph parameter, defined by a composition mechanism for vertex-labeled graphs, which measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic, that includes NP-hard problems) can be solved efficiently for graphs of certified small clique-width. It is widely believed that determining the clique-width of a graph is NP-hard; in spite of considerable efforts, no NP-hardness proof has been found so far. In this paper we show a non-approximability result for restricted form of clique-width, termed r-sequential clique-width, considering only such clique-width constructions where one of any two graphs put together by disjoint union must have r or fewer vertices. In particular, we show that for every positive integer r, the r-sequential clique-width cannot be absolutely approximated in polynomial time unless P=NP, and that given G and k the question of whether the r-sequential clique-width of G is at most k is NP-complete. We show further that this non-approximability result holds even for graphs of a very particular structure: for graphs obtained from cobipartite graphs by replacing edges with induced paths. In part II of this series of papers we use this strengthened result to show that, unless P=NP, there is no polynomial-time absolute approximation algorithm for (unrestricted) clique-width, and that, given a graph G and an integer k, deciding whether the the clique-width G is at most k is NP-complete. This solves a problem that has been open since the introduction of clique-width in the early 1990s.

Abstract:
Clique-width is a graph parameter, defined by a composition mechanism for vertex-labeled graphs, which measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic, that includes NP-hard problems) can be solved efficiently for graphs of certified small clique-width. It is widely believed that determining the clique-width of a graph is NP-hard; in spite of considerable efforts, no NP-hardness proof has been found so far. In this paper we show a non-approximability result for restricted form of clique-width, termed r-sequential clique-width, considering only such clique-width constructions where one of any two graphs put together by disjoint union must have r or fewer vertices. In particular, we show that for every positive integer r, the r-sequential clique-width cannot be absolutely approximated in polynomial time unless P=NP. We show further that this non-approximability result holds even for graphs of a very particular structure: for graphs obtained from cobipartite graphs by replacing edges with induced paths. In part II of this series of papers we use this strengthened result to show that, unless P=NP, there is no polynomial-time absolute approximation algorithm for (unrestricted) clique-width; this solves a problem that has been open since the introduction of clique-width in the early 1990s.