Clique-Width for Hereditary Graph Classes

Abstract

Clique-width is a well-studied graph parameter owing to its use in understanding algorithmic tractability: if the clique-width of a graph class G is bounded by a constant, a wide range of problems that are NP-complete in general can be shown to be polynomial-time solvable on G. For this reason, the boundedness or unboundedness of clique-width has been investigated and determined for many graph classes. We survey these results for hereditary graph classes, which are the graph classes closed under taking induced subgraphs. We then discuss the algorithmic consequences of these results, in particular for the Colouring and Graph Isomorphism problems. We also explain a possible strong connection between results on boundedness of clique-width and on well-quasi-orderability by the induced subgraph relation for hereditary graph classes.

Faster 3-Coloring of Small-Diameter Graphs

“…Prominent examples are perfect graphs [7,18], graphs excluding a certain induced subgraph [17] or minor [11], and intersection graphs of geometric objects [19]. Studying these classes has led to a better understanding of the structure of such graphs [8,9,20,29] and a discovery of numerous exciting algorithmic techniques [2,10,15,16,24]. Let us point out that the property of being hereditary is particularly useful in the construction of recursive algorithms based on branching or the divide & conquer paradigm.…”

Faster 3-coloring of small-diameter graphs

“…Prominent examples are perfect graphs [7,18], graphs excluding a certain induced subgraph [17] or minor [11], and intersection graphs of geometric objects [19]. Studying these classes has led to a better understanding of the structure of such graphs [8,9,20,29] and a discovery of numerous exciting algorithmic techniques [2,10,15,16,24]. Let us point out that the property of being hereditary is particularly useful in the construction of recursive algorithms based on branching or the divide & conquer paradigm.…”