Full-homomorphisms to paths and cycles

A full-homomorphism between a pair of graphs is a vertex mapping that preserves adjacencies and non-adjacencies. For a fixed graph $H$, a full $H$-colouring is a full-homomorphism of $G$ to $H$. A minimal $H$-obstruction is a graph that does not admit a full $H$-colouring, such that every proper induced subgraph of $G$ admits a full $H$-colouring. Feder and Hell proved that for every graph $H$ there is a finite number of minimal $H$-obstructions. We begin this work by describing all minimal obstructions of paths. Then, we study minimal obstructions of regular graphs to propose a description of minimal obstructions of cycles. As a consequence of these results, we observe that for each path $P$ and each cycle $C$, the number of minimal $P$-obstructions and $C$-obstructions is $\mathcal{O}(|V(P)|^2)$ and $\mathcal{O}(|V(C)|^2)$, respectively. Finally, we propose some problems regarding the largest minimal $H$-obstructions, and the number of minimal $H$-obstructions

Matrix partitions of perfect graphs

AbstractGiven a symmetric m by m matrix M over 0,1,*, the M-partition problem asks whether or not an input graph G can be partitioned into m parts corresponding to the rows (and columns) of M so that two distinct vertices from parts i and j (possibly with i=j) are non-adjacent if M(i,j)=0, and adjacent if M(i,j)=1. These matrix partition problems generalize graph colourings and homomorphisms, and arise frequently in the study of perfect graphs; example problems include split graphs, clique and skew cutsets, homogeneous sets, and joins.In this paper we study M-partitions restricted to perfect graphs. We identify a natural class of ‘normal’ matrices M for which M-partitionability of perfect graphs can be characterized by a finite family of forbidden induced subgraphs (and hence admits polynomial time algorithms for perfect graphs). We further classify normal matrices into two classes: for the first class, the size of the forbidden subgraphs is linear in the size of M; for the second class we only prove exponential bounds on the size of forbidden subgraphs. (We exhibit normal matrices of the second class for which linear bounds do not hold.)We present evidence that matrices M which are not normal yield badly behaved M-partition problems: there are polynomial time solvable M-partition problems that do not have finite forbidden subgraph characterizations for perfect graphs. There are M-partition problems that are NP-complete for perfect graphs. There are classes of matrices M for which even proving ‘dichotomy’ of the corresponding M-partition problems for perfect graphs—i.e., proving that these problems are all polynomial or NP-complete—is likely to be difficult

Relations Between Graphs

Given two graphs G and H, we ask under which conditions there is a relation R that generates the edges of H given the structure of graph G. This construction can be seen as a form of multihomomorphism. It generalizes surjective homomorphisms of graphs 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.Comment: accepted by Ars Mathematica Contemporane

Dualities in full homomorphisms

AbstractIn this paper we study dualities of graphs and, more generally, relational structures with respect to full homomorphisms, that is, mappings that are both edge- and non-edge-preserving. The research was motivated, a.o., by results from logic (concerning first order definability) and Constraint Satisfaction Problems. We prove that for any finite set of objects B (finite relational structures) there is a finite duality with B to the left. It appears that the surprising richness of these dualities leads to interesting problems of Ramsey type; this is what we explicitly analyze in the simplest case of graphs

Graph Relations and Constrained Homomorphism Partial Orders

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

The Ranges of K-theoretic Invariants for Nonsimple Graph Algebras

There are many classes of nonsimple graph C*-algebras that are classified by the six-term exact sequence in K-theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained by various classes of graph C*-algebras. To accomplish this, we establish a general method that allows us to form a graph with a given six-term exact sequence of K-groups by splicing together smaller graphs whose C*-algebras realize portions of the six-term exact sequence. As rather immediate consequences, we obtain the first permanence results for extensions of graph C*-algebras. We are hopeful that the results and methods presented here will also prove useful in more general cases, such as situations where the C*-algebras under investigations have more than one ideal and where there are currently no relevant classification theories available.Comment: 40 page