Recognizing Simple-Triangle Graphs by Restricted 2-Chain Subgraph Cover

TL;DR

This paper introduces a simpler polynomial-time recognition algorithm for simple-triangle graphs by utilizing a restricted 2-chain subgraph cover approach, improving understanding and efficiency in recognizing these intersection graphs.

Contribution

It presents a new, simpler polynomial-time recognition algorithm for simple-triangle graphs based on restricted 2-chain subgraph covers, addressing a longstanding open problem.

Findings

Provides a polynomial-time algorithm for recognizing simple-triangle graphs.

Introduces a novel approach using restricted 2-chain subgraph covers.

Simplifies previous recognition methods for these graphs.

Abstract

A simple-triangle graph (also known as a PI graph) is the intersection graph of a family of triangles defined by a point on a horizontal line and an interval on another horizontal line. The recognition problem for simple-triangle graphs was a longstanding open problem, and recently a polynomial-time algorithm has been given [G. B. Mertzios, The Recognition of Simple-Triangle Graphs and of Linear-Interval Orders is Polynomial, SIAM J. Discrete Math., 29(3):1150--1185, 2015]. Along with the approach of this paper, we show a simpler recognition algorithm for simple-triangle graphs. To do this, we provide a polynomial-time algorithm to solve the following problem: Given a bipartite graph $G$ and a set $F$ of edges of $G$, find a 2-chain subgraph cover of $G$ such that one of two chain subgraphs has no edges in $F$.