P6- and triangle-free graphs revisited: structure and bounded clique-width

Andreas Brandstädt,Tilo Klembt,Suhail Mahfud

doi:10.46298/dmtcs.372

Abstract

The Maximum Weight Stable Set (MWS) Problem is one of the fundamental problems on graphs. It is well-known to be NP-complete for triangle-free graphs, and Mosca has shown that it is solvable in polynomial time when restricted to P6- and triangle-free graphs. We give a complete structure analysis of (nonbipartite) P6- and triangle-free graphs which are prime in the sense of modular decomposition. It turns out that the structure of these graphs is extremely simple implying bounded clique-width and thus, efficient algorithms exist for all problems expressible in terms of Monadic Second Order Logic with quantification only over vertex predicates. The problems Vertex Cover, MWS, Maximum Clique, Minimum Dominating Set, Steiner Tree, and Maximum Induced Matching are among them. Our results improve the previous one on the MWS problem by Mosca with respect to structure and time bound but also extends a previous result by Fouquet, Giakoumakis and Vanherpe which have shown that bipartite P6-free graphs have bounded clique-width. Moreover, it covers a result by Randerath, Schiermeyer and Tewes on polynomial time 3-colorability of P6- and triangle-free graphs.

Highlights

Basic problems on graphs such as Vertex Cover and Maximum Weight Stable Set (MWS) which are NPcomplete in general can be solved efficiently for various graph classes
Mosca [21] has shown that MWS can be solved in time O(n4.5) for any P6- and triangle-free graph
We give a complete structure analysis of P6- and triangle-free graphs by showing that such graphs which are not bipartite but prime in the sense of modular decomposition have simple structure which implies bounded clique-width for these graph classes. This leads to the following improvements over previous results:

Summary

Introduction

Basic problems on graphs such as Vertex Cover and Maximum Weight Stable Set (MWS) which are NPcomplete in general can be solved efficiently for various graph classes. For example, the problems Vertex Cover and MWS are NP-complete even for triangle-free graphs [24] but can be solved in polynomial time for bipartite graphs. We give a complete structure analysis of P6- and triangle-free graphs by showing that such graphs which are not bipartite but prime in the sense of modular decomposition have simple structure which implies bounded clique-width for these graph classes. This leads to the following improvements over previous results:. The main result of this paper, namely Theorem 1, gives a complete structure analysis of P6- and triangle-free graphs and implies bounded clique-width of these graphs

Basic Notions and Tools

Time Bound for Robustly Constructing a k-Expression for G

Conclusion