Richard C. Brewster

New bounds for the broadcast domination number of a graph

A dominating broadcast on a graph G = (V, E) is a function f: V → {0, 1, ..., diam G} such that f(v) ≤ e(v) (the eccentricity of v) for all v ∈ V and such that each vertex is within distance f(v) from a vertex v with f(v) > 0. The cost of a broadcast f is σ(f) = Σv∈Vf(v), and the broadcast number λb (G) is the minimum cost of a dominating broadcast.A set X ⊆ V(G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; possibly y = x. The irredundance number ir (G) is the cardinality of a smallest maximal irredundant set of G.We prove the bound λb(G) ≤ 3 ir(G)/2 for any graph G and show that equality is possible for all even values of ir (G). We also consider broadcast domination as an integer programming problem, the dual of which provides a lower bound for λb.

The complexity of signed graph and edge-coloured graph homomorphisms

Abstract We study homomorphism problems of signed graphs from a computational point of view. A signed graph ( G , Σ ) is a graph G where each edge is given a sign, positive or negative; Σ ⊆ E ( G ) denotes the set of negative edges. Thus, ( G , Σ ) is a 2 -edge-coloured graph with the property that the edge-colours, { + , − } , form a group under multiplication. Central to the study of signed graphs is the operation of switching at a vertex, that results in changing the sign of each incident edge. We study two types of homomorphisms of a signed graph ( G , Σ ) to a signed graph ( H , Π ) : ec-homomorphisms and s-homomorphisms. Each is a standard graph homomorphism of G to H with some additional constraint. In the former, edge-signs are preserved. In the latter, edge-signs are preserved after the switching operation has been applied to a subset of vertices of G . We prove a dichotomy theorem for s-homomorphism problems for a large class of (fixed) target signed graphs ( H , Π ) . Specifically, as long as ( H , Π ) does not contain a negative (respectively a positive) loop, the problem is polynomial-time solvable if the core of ( H , Π ) has at most two edges, and is NP-complete otherwise. (Note that this covers all simple signed graphs.) The same dichotomy holds if ( H , Π ) has no negative digons, and we conjecture that it holds always. In our proofs, we reduce s-homomorphism problems to certain ec-homomorphism problems, for which we are able to show a dichotomy. In contrast, we prove that a dichotomy theorem for ec-homomorphism problems (even when restricted to bipartite target signed graphs) would settle the dichotomy conjecture of Feder and Vardi.

Edge-switching homomorphisms of edge-coloured graphs

An edge-coloured graph G is a vertex set V(G) together with m edge sets distinguished by m colours. Let @p be a permutation on {1,2,...,m}. We define a switching operation consisting of @p acting on the edge colours similar to Seidel switching, to switching classes as studied by Babai and Cameron, and to the pushing operation studied by Klostermeyer and MacGillivray. An edge-coloured graph G is @p-permutably homomorphic (respectively @p-permutably isomorphic) to an edge-coloured graph H if some sequence of switches on G produces an edge-coloured graph homomorphic (respectively isomorphic) to H. We study the @p-homomorphism and @p-isomorphism operations, relating them to homomorphisms and isomorphisms of a constructed edge-coloured graph that we introduce called a colour switching graph. Finally, we identify those edge-coloured graphs H with the property that G is homomorphic to H if and only if any switch of G is homomorphic to H. It turns out that such an H is precisely a colour switching graph. As a corollary to our work, we settle an open problem of Klostermeyer and MacGillivray.

Mixing Homomorphisms, Recolorings, and Extending Circular Precolorings

This work brings together ideas of mixing graph colorings, discrete homotopy, and precoloring extension. A particular focus is circular colorings. We prove that all the k,q-colorings of a graph G can be obtained by successively recoloring a single vertex provided k/qi¾?2colG along the lines of Cereceda, van den Heuvel, and Johnsons result for k-colorings. We give various bounds for such mixing results and discuss their sharpness, including cases where the bounds for circular and classical colorings coincide. As a corollary, we obtain an Albertson-type extension theorem for k,q-precolorings of circular cliques. Such a result was first conjectured by Albertson and West. General results on homomorphism mixing are presented, including a characterization of graphs G for which the endomorphism monoid can be generated through the mixing process. As in similar work of Brightwell and Winkler, the concept of dismantlability plays a key role.

On the hamiltonicity of line graphs of locally finite, 6-edge-connected graphs

The topological approach to the study of infinite graphs of Diestel and KUhn has enabled several results on Hamilton cycles in finite graphs to be extended to locally finite graphs. We consider the result that the line graph of a finite 4-edge-connected graph is hamiltonian. We prove a weaker version of this result for infinite graphs: The line graph of locally finite, 6-edge-connected graph with a finite number of ends, each of which is thin, is hamiltonian.

On the restricted homomorphism problem

The restricted homomorphism problemRHP(H,Y) asks: given an input digraph G and a homomorphism g:G->Y, does there exist a homomorphism f:G->H? We prove that if H is hereditarily hard and Y@?H, then RHP(H,Y) is NP-complete. Since non-bipartite graphs are hereditarily hard, this settles in the affirmative a conjecture of Hell and Nesetril.

The recognition of bound quivers using edge-coloured homomorphisms

A bound quiver is a digraph together with a collection of specified directed walks. Given an undirected graph G, and a collection of walks I, the bound quiver recognition problem asks: Is there an orientation of G such that each walk in I is directed? We present a polynomial time algorithm for this problem, and a structural characterization of which trees can be oriented as bound quivers. These results are developed using homomorphisms of edge-coloured graphs. Our work includes a classification of the computational complexity of edge-coloured homomorphism problems where the target is of order at most three.

A complexity dichotomy for signed H-colouring.

Abstract Verifying a conjecture of Brewster, Foucaud, Hell and Naserasr, we show that signed ( H , Π ) -colouring is NP-complete for any signed graph ( H , Π ) whose s-core has at least 3 edges.

Recolouring reflexive digraphs

Abstract Given digraphs G and H , the colouring graph Col ( G , H ) has as its vertices all homomorphism of G to H . There is an arc ϕ → ϕ ′ between two homomorphisms if they differ on exactly one vertex v , and if v has a loop we also require ϕ ( v ) → ϕ ′ ( v ) . The recolouring problem asks if there is a path in Col ( G , H ) between given homomorphisms ϕ and ψ . We examine this problem in the case where G is a digraph and H is a reflexive, digraph cycle. We show that for a reflexive digraph cycle H and a reflexive digraph G , the problem of determining whether there is a path between two maps in Col ( G , H ) can be solved in time polynomial in G . When G is not reflexive, we show the same except for certain digraph 4-cycles H .

Factors with Multiple Degree Constraints in Graphs

For a graph