Shannon L. Fitzpatrick

The strong isometric dimension of finite reflexive graphs

The strong isometric dimension of a reflexive graph is related to its injective hull: both deal with embedding reflexive graphs in the strong product of paths. We give several upper and lower bounds for the strong isometric dimension of general graphs; the exact strong isometric dimension for cycles and hypercubes; and the isometric dimension for trees is found to within a factor of two.

A deterministic version of the game of zombies and survivors on graphs

We consider a variant of the pursuit-evasion game Cops and Robber, called Zombies and Survivors. The zombies, being of limited intelligence, have a very simple objective at each round: move closer to a survivor. The zombies capture a survivor if one of the zombies moves onto the same vertex as a survivor. The survivors objective is to avoid capture for as long as possible, hopefully indefinitely. Because there may be multiple geodesics, or shortest paths, joining a zombie and its nearest survivor, the game can be considered from a probabilistic or deterministic approach. In this paper, we consider a deterministic approach to the game. In particular, we consider the worst case for the survivors; whenever the zombies have more than one possible move, they choose one that works to their advantage. This includes choice of initial position, and choosing which geodesic to move along if more than one is available. In other words, the zombies play intelligently, subject to the constraint that each zombie must move along a geodesic between itself and the nearest survivor. The zombie number of a graph G is the minimum number of zombies required to capture the survivor on G . We determine the zombie number for various graphs, examine the relationship between the zombie number and cop number of a graph, and describe some distinctions from Cops and Robber.

Lower Bounds from Tile Covers for the Channel Assignment Problem

A method to generate lower bounds for the channel assignment problem is given. The method is based on the reduction of the channel assignment problem to a problem of covering the demand in a cellular network by preassigned blocks of cells called tiles. This tile cover approach is applied to networks with a cosite constraint and two different constraints between cells. A complete family of lower bounds is obtained, which include a number of new bounds that improve or include almost all known clique bounds. When applied to an example from the literature, the new bounds give better results.

Distributive online channel assignment for hexagonal cellular networks with constraints

In cellular networks, channels must be assigned to call requests so that interference constraints are respected and bandwidth is minimized. The number of call requests per cell is continually changing, making channel assignment naturally an online problem. We describe two new online channel assignment algorithms for networks based on a regular hexagonal layout of cells, where interference levels depend only on the distance between cells. Such networks can be modeled by so-called hexagon graphs. Our model incorporates different separation constraints, prescribed minimal differences between channels assigned to cells within a certain distance of each other. The algorithms presented are the first to take into account separation constraints between non-adjacent cells in this type of layout. The algorithms are distributed in nature: each cell server will need only a limited exchange of information with cells in its proximity to make decisions on its channel assignment.

Edge-critical cops and robber in planar graphs ☆

Abstract The problem is to determine the number of ‘cops’ needed to capture a ‘robber’ in a game in which the cops always know the location of the robber, and the cops and robber move alternately along edges of a reflexive graph. The cops capture the robber if one of them occupies the same vertex as the robber at any time in the game. A cop-win graph is one in which a single cop has a winning strategy. A graph is cop-win edge-critical with respect to edge addition (respectively, deletion) when the original graph is not cop-win, but the addition (deletion) of any edge results in a cop-win graph. In this paper, edge-critical planar graphs are characterized.

Edge critical cops and robber

We consider edge critical graphs when playing cops and robber. Specifically, we look at those graphs whose copnumbers change from one to two when any edge is added, deleted, subdivided or contracted. We characterize all such sets, showing that they are empty, trees, all 2-edge-connected graphs and empty, respectively. We also consider those graphs which change from copnumber two to one when any edge is added, and give a characterization in the k-regular case.

A note on the Grundy number and graph products

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.

Well paired-dominated graphs

A paired-dominating set is a dominating set whose induced subgraph contains at least one perfect matching. This could model the situation of guards or police where each has a partner or backup. We are interested in those where all “minimal” paired-dominating sets are the same cardinality. In this case, we consider “minimal” to be with respect to the pairings. That is, the removal of any two vertices paired under the matching results in a set that is not dominating. We give a structural characterization of all such graphs with girth at least eight.

The game of Cops and Robber on circulant graphs

We examine the game of Cops and Robber on circulant graphs, and determine the copnumbers of all circulant graphs of degree at most four. We then look at wreath products, and show how they can be used to determine the copnumbers for additional classes of circulant graphs. Finally, we show how expressing a circulant graph as a wreath product relates to dismantling that graph via corners and open corners.