On fundamental results for pushable homomorphisms of oriented graphs

TL;DR

This paper explores pushable homomorphisms of oriented graphs, establishing fundamental properties, connections to signed graphs and coloring, and providing polynomial-time decision algorithms and NP-completeness results.

Contribution

It introduces a canonical definition of pushable homomorphisms, characterizes push equivalence of orientations, and links oriented graph homomorphisms to signed graphs and graph coloring.

Findings

Polynomial-time algorithm for push equivalence of orientations

Canonical definition of pushable homomorphism

NP-completeness of pushable homomorphism to directed odd cycles

Abstract

This article deals with homomorphisms of oriented graphs with respect to push equivalence. Here homomorphisms refer to arc preserving vertex mappings, and push equivalence refers to the equivalence class of orientations of a graph $G$ those can be obtained from one another by reversing arcs of an edge cut. We study and prove some fundamental properties of pushable homomorphisms, and establish its connections to homomorphisms of signed graphs and graph coloring. To list a few highlights of this work: $\bullet$ We characterize orientations of a graph up to push equivalence and show that it is possible to decide whether they are equivalent or not in polynomial time. $\bullet$ We give a canonical definition of pushable homomorphism - this answers a natural open question. $\bullet$ We build a one-to-one correspondence between the equivalence classes of oriented and signed bipartite graphs. Thus, it is possible to translate a number of important results directly from the theory of signed graphs to oriented graphs. In particular, we show that pushable homomorphisms of bipartite graphs capture the entire theory of graph coloring as a subcase. $\bullet$ Given a graph $G$, we build a gadget oriented graph $\overrightarrow{G}^{(k)}$ which admits a pushable homomorphism to a directed odd cycle of length $(2k+1)$ if and only if $G$ admits a $(2k+1)$-coloring. We also show that it is NP-complete to determine whether an oriented (sparse) graph admits a pushable homomorphism to a directed odd cycle or not.