On Computational Aspects of Cores of Ordered Graphs

TL;DR

This paper explores the computational complexity of core-related problems in ordered graphs, revealing polynomial-time solvability for some tasks and NP-hardness for core recognition, highlighting differences from unordered graphs.

Contribution

It introduces complexity results for core problems in ordered graphs, including polynomial algorithms and NP-hardness proofs, contrasting with unordered graph cases.

Findings

Retraction problem is solvable in polynomial time.

Deciding if an ordered graph is a core is NP-hard.

Distinguishing graphs by core size is NP-hard and W[1]-hard.

Abstract

An ordered graph is a graph enhanced with a linear order on the vertex set. An ordered graph is a core if it does not have an order-preserving homomorphism to a proper subgraph. We say that $H$ is the core of $G$ if (i) $H$ is a core, (ii) $H$ is a subgraph of $G$, and (iii) $G$ admits an order-preserving homomorphism to $H$. We study complexity aspects of several problems related to the cores of ordered graphs. Interestingly, they exhibit a different behavior than their unordered counterparts. We show that the retraction problem, i.e., deciding whether a given graph admits an ordered-preserving homomorphism to its specific subgraph, can be solved in polynomial time. On the other hand, it is \NP-hard to decide whether a given ordered graph is a core. In fact, we show that it is even \NP-hard to distinguish graphs $G$ whose core is largest possible (i.e., if $G$ is a core) from those, whose core is the smallest possible, i.e., its size is equal to the ordered chromatic number of $G$. The problem is even \wone-hard with respect to the latter parameter.