Recognizing Berge Graphs

Abstract

A graph is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. In this paper we give an algorithm to test if a graph G is Berge, with running time O(|V (G)| 9 ). This is independent of the recent proof of the strong perfect graph conjecture. (2000): 05C17 Mathematics Subject Classification

Largest Chordal and Interval Subgraphs Faster than $$2^n$$ 2 n

“…We have two concrete examples of hereditary, polynomial-time recognizable, and CMSO 1 -definable graph classes for which we do not know any algorithm faster than 2 n : -Perfect graphs can be defined as graphs which contain neither an odd hole, nor an odd anti-hole; this result is known as the Strong Perfect Graph Theorem [5]. Perfect graphs are hereditary, polynomial-time recognizable [4], and containing an odd hole or an odd anti-hole can be easily expressed in CMSO 1 . -Strongly chordal graphs are chordal graphs that moreover exclude -suns (graphs with 2 vertices partitioned in two sets W = {w 1 , w 2 , .…”

Even-hole-free graphs: A survey

“…Note that the condition that nodes of P i ∪ P j , i = j, must induce a hole, implies that all paths of a 3P C(·, ·) have length greater than one, and at most one path of a 3P C(∆, ·) has length one. 3P C(·, ·)'s are also known as thetas (as in [8]), 3P C(∆, ∆)'s are also known as prisms (as in [8]), and 3P C(∆, ·)'s are also known as pyramids (as in [7]). …”

On Parameterized Path and Chordless Path Problems

“…A hole is even (odd) if it has even (odd) length. In the last time problems concerning holes have received much attention as they are related to the Strong Perfect Graph Theorem ("A graph is perfect if it contains neither an odd hole nor the complement of an odd hole"), which has been proven recently [6,8]. We mention some results and open problems: It is not known whether there is a polynomial time algorithm deciding if a graph has an odd hole, while the questions whether a graph contains a hole and whether it contains an even hole are solvable in polynomial time (cf.…”

“…Recently, the search for algorithms detecting chordless cycles (of odd length ≥ 5) has received much attention due to its relationship to Berge graphs and to the Strong Perfect Graph Theorem (cf. [6,8,10,11,26]). …”