Complexity Aspects of Homomorphisms of Ordered Graphs
Michal Čertík, Andreas Emil Feldmann, Jaroslav Nešetřil, Paweł Rzążewski
TL;DR
The paper investigates the complexity of homomorphisms between ordered graphs, introducing a rigorous reduction from unordered structures to ordered graphs via ordered bipartite constructions. It establishes that fixed-$H$ coloring problems on ordered graphs are polynomial-time solvable, while the general Hom$_{<}$ problem is NP-complete and parameterized variants are W[1]-hard under ETH; it also places the problem in XP when parameterized by $|V(H)|$. The work further identifies tractable cases, notably $k$-shifted cliques and graphs with small generalized pathwidth, and provides algorithmic frameworks (including dynamic programming) to solve Hom$_{<}$ in these regimes. Overall, it delineates a nuanced landscape of complexity for ordered-graph homomorphisms, connecting reductions, fixed-parameter tractability, and structural graph properties. This advances understanding of how ordering constraints influence classical graph homomorphism problems and their computational boundaries.
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.