Efficient computation of oriented vertex and arc colorings of special digraphs

TL;DR

This paper presents efficient algorithms for computing the oriented chromatic number and index of special classes of digraphs, specifically edge series-parallel and minimal series-parallel digraphs, with tight bounds and linear time solutions.

Contribution

It introduces the first linear time algorithms for these coloring problems on these classes of digraphs and establishes tight upper bounds for their oriented chromatic parameters.

Findings

Tight upper bounds for oriented chromatic number and index.

Linear time algorithms for computing these parameters.

Application of results to special classes of digraphs.

Abstract

In this paper we study the oriented vertex and arc coloring problem on edge series-parallel digraphs (esp-digraphs) which are related to the well known series-parallel graphs. Series-parallel graphs are graphs with two distinguished vertices called terminals, formed recursively by parallel and series composition. These graphs have applications in modeling series and parallel electric circuits and also play an important role in theoretical computer science. The oriented class of series-parallel digraphs is recursively defined from pairs of vertices connected by a single arc and applying the parallel and series composition, which leads to specific orientations of undirected series-parallel graphs. Further we consider the line digraphs of edge series-parallel digraphs, which are known as minimal series-parallel digraphs (msp-digraphs). We show tight upper bounds for the oriented chromatic number and the oriented chromatic index of edge series-parallel digraphs and minimal series-parallel digraphs. Furthermore, we introduce first linear time solutions for computing the oriented chromatic number of edge series-parallel digraphs and the oriented chromatic index of minimal series-parallel digraphs.