Abstract

Bi-arc graphs generalize (reflexive) interval graphs and those (irreflexive) bipartite graphs whose complements are circular arc graphs. They are relevant for the so-called list homomorphism problem: when H is a bi-arc graph, the problem is polynomial time solvable, otherwise it is NP-complete. Bi-arc graphs have a forbidden structure characterization, and can be recognized in polynomial time. More importantly for this paper, bi-arc graphs can be characterized by the existence of a conservative majority function. (This function plays an important role in proving the correctness of a polynomial time list homomorphism algorithm.)

The forbidden structure theorem for bi-arc graphs is quite complex, and the existence of a conservative majority function is proved without giving an explicit description of it.

In this note we focus on bi-arc graphs that are trees (with loops allowed). We describe the structure of bi-arc trees, and give a simple forbidden subtree characterization. Based on this structure theorem, we are able to explicitly describe the conservative majority functions.

Keywords: Homomorphism; List homomorphism; Bi-arc graph; Bi-arc tree; Circular arc graph; Majority function; Forbidden subgraph characterization

Article Outline

1. Introduction

2. The structure theorem

3. The conservative majority functions

References