Acyclicity in Edge-Colored Graphs

TL;DR

This paper introduces five types of proper-colored (PC) acyclicity in edge-colored graphs, analyzing their properties, computational complexity, and implications for path problems, revealing a surprising NP-hardness result for one type.

Contribution

It defines and studies five new PC acyclicity types, establishing their recognition complexity and exploring their impact on disjoint path and vertex cover problems.

Findings

Recognition of types 1-3 is polynomial, type 4 is NP-hard, type 5 is polynomial.

NP-hardness of recognizing type 4 graphs even with only two colors.

Analysis of tractability boundaries for PC path and vertex cover problems.

Abstract

A walk $W$ in edge-colored graphs is called properly colored (PC) if every pair of consecutive edges in $W$ is of different color. We introduce and study five types of PC acyclicity in edge-colored graphs such that graphs of PC acyclicity of type $i$ is a proper superset of graphs of acyclicity of type $i+1$, $i=1,2,3,4.$ The first three types are equivalent to the absence of PC cycles, PC trails, and PC walks, respectively. While graphs of types 1, 2 and 3 can be recognized in polynomial time, the problem of recognizing graphs of type 4 is, somewhat surprisingly, NP-hard even for 2-edge-colored graphs (i.e., when only two colors are used). The same problem with respect to type 5 is polynomial-time solvable for all edge-colored graphs. Using the five types, we investigate the border between intractability and tractability for the problems of finding the maximum number of internally vertex disjoint PC paths between two vertices and the minimum number of vertices to meet all PC paths between two vertices.