Convex obstacle numbers of outerplanar graphs and bipartite permutation graphs

TL;DR

This paper investigates the disjoint convex obstacle number for specific graph classes, establishing upper bounds for outerplanar and bipartite permutation graphs and answering a previously open question.

Contribution

It proves that outerplanar graphs have a maximum disjoint convex obstacle number of 5 and bipartite permutation graphs at most 4, providing new bounds and resolving an open problem.

Findings

Outerplanar graphs have obstacle number at most 5.

Bipartite permutation graphs have obstacle number at most 4.

Lower bound of 4 for outerplanar graphs established.

Abstract

The disjoint convex obstacle number of a graph G is the smallest number h such that there is a set of h pairwise disjoint convex polygons (obstacles) and a set of n points in the plane (corresponding to V(G)) so that a vertex pair uv is an edge if and only if the corresponding segment uv does not meet any obstacle. We show that the disjoint convex obstacle number of an outerplanar graph is always at most 5, and of a bipartite permutation graph at most 4. The former answers a question raised by Alpert, Koch, and Laison. We complement the upper bound for outerplanar graphs with the lower bound of 4.