[Paper Review] Graph Relations and Constrained Homomorphism Partial Orders

Yangjing Long|arXiv (Cornell University)|Apr 21, 2014

Advanced Graph Theory Research4 references3 citations

TL;DR

This paper introduces a novel framework for constrained graph homomorphisms by defining graph relations and R-homomorphisms, which generalize surjective and locally constrained homomorphisms. It establishes R-cores and R-cocores as unique up to isomorphism and computable in polynomial time, and proves universality of the resulting partial orders, including for line graphs of bounded-degree graphs and oriented cycles.

ABSTRACT

We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective homomorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives from relations between graphs and is related to multihomomorphisms. This gives a generalization of surjective homomorphisms and naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time. The theory of the graph homomorphism order is well developed, and from it we consider analogous notions defined for orders induced by constrained homomorphisms. We identify corresponding cores, prove or disprove universality, characterize gaps and dualities. We give a new and significantly easier proof of the universality of the homomorphism order by showing that even the class of oriented cycles is universal. We provide a systematic approach to simplify the proofs of several earlier results in this area. We explore in greater detail locally injective homomorphisms on connected graphs, characterize gaps and show universality. We also prove that for every $d\geq 3$ the homomorphism order on the class of line graphs of graphs with maximum degree $d$ is universal.

Motivation & Objective

• To generalize classical graph homomorphism orders by introducing constrained variants such as embeddings, monomorphisms, and surjective homomorphisms.
• To define a new class of homomorphisms via graph relations (R-homomorphisms), extending surjective and locally constrained homomorphisms.
• To establish the existence and uniqueness of R-cores and R-cocores up to isomorphism, and to show they are computable in polynomial time.
• To investigate universality, gaps, and dualities in the partial orders induced by constrained homomorphisms.
• To provide a simplified proof of the universality of the standard homomorphism order using oriented cycles.

Proposed method

• Define R-homomorphisms as a generalization of surjective homomorphisms through relational constraints between graphs.
• Introduce R-retractions and use them to define R-cores and R-cocores as canonical representatives of graph classes under R-homomorphism equivalence.
• Apply weak relational composition and relational equivalence to analyze the structure of the resulting partial orders.
• Use the theory of past-finite and future-finite orders to characterize gaps and dualities in constrained homomorphism orders.
• Prove universality by showing that the class of oriented cycles induces a universal partial order.
• Leverage line graph constructions and degree-bounded graph families to extend universality results to line graphs of maximum degree d.

Experimental results

Research questions

• RQ1 What is the structure of the partial order induced by R-homomorphisms, and how do R-cores and R-cocores relate to it?
• RQ2 Are R-cores and R-cocores unique up to isomorphism, and can they be computed in polynomial time?
• RQ3 Does the partial order of constrained homomorphisms (e.g., locally injective, surjective) exhibit universality?
• RQ4 Can the universality of the standard homomorphism order be proven more simply using R-homomorphisms or oriented cycles?
• RQ5 What are the conditions under which dualities and gaps exist in R-homomorphism orders?

Key findings

• R-cores and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time.
• The class of oriented cycles induces a universal partial order, providing a significantly simpler proof of the universality of the standard homomorphism order.
• For every d ≥ 3, the homomorphism order on the class of line graphs of graphs with maximum degree d is universal.
• The paper provides a systematic simplification of earlier proofs in the theory of constrained homomorphisms and dualities.
• Gaps in locally injective homomorphism orders are characterized, and dualities are identified in future-finite orders.
• The theory of R-homomorphisms generalizes surjective and locally constrained homomorphisms, offering a unified framework for studying constrained graph mappings.