Elsevier

Discrete Mathematics

Volume 307, Issues 3–5, 6 February 2007, Pages 393-401
Discrete Mathematics

The structure of bi-arc trees

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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

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