Complexity Aspects of Homomorphisms of Ordered Graphs

TL;DR

This paper explores the computational complexity of homomorphisms in ordered graphs, establishing NP-completeness, parameterized hardness, and polynomial-time solvable classes, with implications for graph coloring and structure analysis.

Contribution

It introduces complexity classifications for ordered graph homomorphisms, including NP-completeness results, reductions from unordered structures, and identifies tractable subclasses.

Findings

Homomorphism problems are NP-complete for ordered graphs.

The problem is W[1]-hard when parameterized by the size of the image.

Certain classes of ordered graphs admit polynomial-time algorithms.

Abstract

We examine ordered graphs, defined as graphs with linearly ordered vertices, from the perspective of homomorphisms (and colorings) and their complexities. We demonstrate the corresponding computational and parameterized complexities, along with algorithms associated with related problems. These questions are interesting, and we show that numerous problems lead to various complexities. The reduction from homomorphisms of unordered structures to homomorphisms of ordered graphs is proved, achieved with the use of ordered bipartite graphs. We then determine the NP-completeness of the problem of finding ordered homomorphisms of ordered graphs and the XP and W[1]-hard nature of this problem parameterized by the number of vertices of the image ordered graph. Classes of ordered graphs for which this problem can be solved in polynomial time are also presented.