Stephen Finbow

A lower bound of the surviving rate of a planar graph with girth at least seven

Let G be a connected graph with n≥2 vertices. Suppose that a fire breaks out at a vertex v of G. A firefighter starts to protect vertices. At each time interval, the firefighter protects one vertex not yet on fire. At the end of each time interval, the fire spreads to all the unprotected vertices that have a neighbor on fire. Let sn(v) denote the maximum number of vertices in G that the firefighter can save when a fire breaks out at vertex v. The surviving rate ρ(G) of G is defined to be ∑v∈V(G)sn(v)/n2, which is the average proportion of saved vertices.In this paper, we show that if G is a planar graph with n≥2 vertices and having girth at least 7, then

The Robber Strikes Back

\rho(G)>\frac{1}{301}

Triangulations and equality in the domination chain

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A simple method of computing the catch time

We consider the new game of Cops and Attacking Robbers, which is identical to the usual Cops and Robbers game except that if the robber moves to a vertex containing a single cop, then that cop is removed from the game. We study the minimum number of cops needed to capture a robber on a graph G, written cc(G). We give bounds on cc(G) in terms of the cop number of G in the classes of bipartite graphs and diameter two, K 1, m -free graphs.

A note on the eternal dominating set problem

In this paper we give a characterization of triangulations where all minimal dominating sets have the same cardinality and a characterization of triangulations where all six domination parameters are the same.

A note on the Grundy number and graph products

We describe a simple method for computing the maximum length of the game cop and robber, assuming optimal play for both sides.

An improved bound on parity vertex colourings of outerplane graphs

We consider the “all guards move” model for the eternal dominating set problem. A set of guards form a dominating set on a graph and at the beginning of each round, a vertex not in the dominating set is attacked. To defend against the attack, the guards move (each guard either passes or moves to a neighboring vertex) to form a dominating set that includes the attacked vertex. The minimum number of guards required to defend against any sequence of attacks is the “eternal domination number” of the graph. In 2005, it was conjectured [Goddard et al. (J. Combin. Math. Combin. Comput. 52:169–180, 2005)] there would be no advantage to allow multiple guards to occupy the same vertex during a round. We show this is, in fact, false. We also describe algorithms to determine the eternal domination number for both models for eternal domination and examine the related combinatorial game, which makes use of the reduced canonical form of games.

The Firefighter Problem: a survey of results, directions and questions.

A proper colouring is referred to as a Grundy colouring, or first-fit colouring if every vertex has a neighbour from each of the colour classes lower than its own. The Grundy number of a graph is the maximum k (number of colours) such that a Grundy colouring exists.In this note, we determine lower and upper bounds for the Grundy number of strong products of graphs, which lead to exact values for the product of some graph classes. We also provide an upper bound on the Grundy number of the strong product of n paths of length 2, which generalizes to an upper bound on the Grundy number of the strong product of n stars.

The surviving rate of an infected network

Abstract A parity vertex colouring of a 2-connected plane graph G is a proper vertex colouring such that for each face f and colour i , either zero or an odd number of vertices incident with f are coloured i . The parity chromatic number χ p ( G ) of G is the smallest number of colours used in a parity vertex colouring of G . In this paper, we improve a result of Czap by showing that every 2-connected outerplane graph G , with two exceptions, has χ p ( G ) ≤ 9 . In addition, we characterize the 2-connected outerplane graphs G with χ p ( G ) = 2 and those which are bipartite and have χ p ( G ) = 8 .