Gary MacGillivray

The firefighter problem for graphs of maximum degree three

We show that the firefighter problem is NP-complete for trees of maximum degree three, but in P for graphs of maximum degree three if the fire breaks out at a vertex of degree at most two.

A note on k-cop, l-robber games on graphs

We give an algorithmic characterisation of finite cop-win digraphs. The case of k>1 cops and k>=l>=1 robbers is then reduced to the one cop case. Similar characterisations are also possible in many situations where the movements of the cops and/or the robbers are somehow restricted.

The complexity of colouring by semicomplete digraphs

The following problem, known as the H-colouring problem, is studied. An H-colouring of a directed graph D is a mapping

Vertex domination-critical graphs

f:V( D ) \to V( H )

Domination numbers of planar graphs

such that

The firefighter problem for cubic graphs

( f( x ),f( y ) )

Homomorphisms and oriented colorings of equivalence classes of oriented graphs

is an edge of H whenever

CO-irredundant Ramsey numbers for graphs

( x,y )

Characterizations of k-copwin graphs

is an edge of D. The H-colouring problem is the following. Instance: A directed graph D. Question: Does there exist an H-colouring of D? In this paper it is shown that for semicomplete digraphs T the T-colouring problem is NP-complete when T has more than one directed cycle, and polynomially decidable otherwise.

Efficient domination in circulant graphs

A graph G is vertex domination-critical if for any vertex v of G the domination number of G - v is less than the domination number of G. If such a graph G has domination number γ, it is called γ-critical. Brigham et al. studied γ-critical graphs and posed the following questions: (1) If G is a γ-critical graph, is |V| ≥ (δ + 1)(γ - 1) + 1?(2) If a γ-critical graph G has (Δ + 1)(γ - 1) + 1 vertices, is G regular? (3) Does i = γ for all γ-critical graphs? (4) Let d be the diameter of the γ-critical graph G. Does d ≤ 2(γ - 1) always hold? We show that the first and third questions have a negative answer and the others have a positive answer.