A recognition algorithm for simple-triangle graphs

TL;DR

This paper introduces a more efficient algorithm for recognizing simple-triangle graphs, reducing the time complexity from quadratic to linearithmic, and explores related orientations in cocomparability graphs.

Contribution

It presents a new recognition algorithm for simple-triangle graphs with improved time complexity and characterizes vertex orderings involving alternating and transitive orientations.

Findings

Recognition algorithm runs in O(nm) time.

Alternating orientation can be obtained in O(nm) for cocomparability graphs.

Deciding if a graph has an alternating and acyclic orientation is NP-complete.

Abstract

A simple-triangle graph is the intersection graph of triangles that are defined by a point on a horizontal line and an interval on another horizontal line. The time complexity of the recognition problem for simple-triangle graphs was a longstanding open problem, which was recently settled. This paper provides a new recognition algorithm for simple-triangle graphs to improve the time bound from $O(n^2 \overline{m})$ to $O(nm)$, where $n$, $m$, and $\overline{m}$ are the number of vertices, edges, and non-edges of the graph, respectively. The algorithm uses the vertex ordering characterization that a graph is a simple-triangle graph if and only if there is a linear ordering of the vertices containing both an alternating orientation of the graph and a transitive orientation of the complement of the graph. We also show, as a byproduct, that an alternating orientation can be obtained in $O(nm)$ time for cocomparability graphs, and it is NP-complete to decide whether a graph has an orientation that is alternating and acyclic.