On Computational Aspects of Cores of Ordered Graphs

TL;DR

This work investigates the computational complexity of cores in ordered graphs, focusing on order-preserving homomorphisms and retractions. It shows a sharp contrast with unordered graphs: the RET_< retraction problem is solvable in polynomial time via a 2-SAT reduction, while deciding if an ordered graph is a core is NP-hard and even W[1]-hard when parameterized by core size. The authors prove NP-hardness and related results through intricate gadget constructions from ONE-IN-THREE SAT and analyze sliced-subgraph variants SLICE_<gh and SUB_<tu to highlight hardness boundaries. They also establish that every finite ordered graph is homomorphically equivalent to a unique ordered core, providing a canonical target for reductions and a bridge between structural graph theory and computational complexity. Collectively, the paper delineates a nuanced landscape of tractability and hardness for ordered-graph cores, with implications for CSPs and structural graph normalization.

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.

Figures (2)

• Figure 1: Ordered Homomorphism $f$ and Independent Intervals.
• Figure 2: Ordered Hypergraph $G$ from Theorem \ref{['thm:corehyper']}.

Theorems & Definitions (25)

• theorem 1.1
• proof
• theorem 1.2
• proof
• theorem 1.3
• proof
• corollary thmcountercorollary
• proof
• theorem 1.4
• proof