This is an account for the combinatorially minded reader of various categories of directed and undirected graphs, and their analogies with the category of sets. As an application, the endomorphisms of a graph are in this context not only composable, giving a monoid structure, but also have a notion of adjacency, so that the set of endomorphisms is both a monoid and a graph. We extend Shrimpton’...

It is proved that a split graph is an absolute retract of split graphs if and only if a partition of its vertex set into a stable set and a complete set is unique or it is a complete split graph. Three equivalent conditions for a split graph to be an absolute retract of the class of all graphs are given. It is finally shown that a reflexive split graph G is an absolute retract of reflexive spli...

Let H be a graph and k ≥ 3. A near-unanimity function of arity k is a mapping g from the k-tuples over V (H) to V (H) such that g(x1, x2, . . . , xk) is adjacent to g(x ′ 1, x ′ 2, . . . , x ′ k) whenever xix ′ i ∈ E(H) for each i = 1, 2, . . . , k, and g(x1, x2, . . . , xk) = a whenever at least k − 1 of the xi’s equal a. Feder and Vardi proved that, if a graph H admits a near-unanimity functi...

A first order structure M with universe M is atomic compact if every system of atomic formulas with parameters in M is satisfiable in M provided each of its finite subsystems is. We consider atomic compactness for the class of reflexive (symmetric) graphs. In particular, we investigate the extent to which “sparse” graphs (i.e. graphs with “few” vertices of “high” degree) are compact with respec...

In this paper, we consider terminal reflexive polytopes arising from finite directed graphs and study the problem of deciding which directed graphs yield smooth Fano polytopes. We show that any centrally symmetric or pseudosymmetric smooth Fano polytopes can be obtained from directed graphs. Moreover, by using directed graphs, we provide new examples of smooth Fano polytopes whose corresponding...

For the moment, we outline how to characterise undirected, irreflexive graphs that admit a near-unanimity operation of arity at most k using the notion of dual of a tree; the same is done for reflexive graphs, and for directed bipartite graphs. We hope to complete this by showing that the generating sets for the varieties in question contain only chordal extensible graphs.

We present a simple polynomial-time algorithm that recognises reflexive, symmetric graphs admitting a near-unanimity operation. Several other characterisations of these graphs are also presented.  2004 Elsevier Inc. All rights reserved.

Let H be a fixed graph. We introduce the following list homomorphism problem: Given an input graph G and for each vertex v of G a ``list'' L(v) V(H), decide whether or not there is a homomorphism f : G H such that f (v) # L(v) for each v #V(G). We discuss this problem primarily in the context of reflexive graphs, i.e., graphs in which each vertex has a loop. We give a polynomial time algorithm ...

For a graph H, the H-recolouring problem \({\text {Recol}}(H)\) asks, for two given homomorphisms from G to if one can get between them by sequence of H in which consecutive differ on only vertex. We show that, and are reflexive is triangle-free, then this be solved polynomial time. This shows, at same time, that closely related H-reconfiguration {Recon}}(H)\) deciding whether component Hom-gra...

Given a 3-colorable graph X , the 3-coloring complex B ( ) is whose vertices are all independent sets which occur as color classes in some of . Two C D ∈ V joined by an edge if and appear together The 3-colorable. Graphs for isomorphic to termed reflexive graphs. In this paper, we consider 3-edge-colorings cubic graphs allow half-edges. Then complexes their line main result paper surprising out...