Margaret-Ellen Messinger

Clean the graph before you draw it

We prove a relationship between the Cleaning problem and the Balanced Vertex-Ordering problem, namely that the minimum total imbalance of a graph equals twice the brush number of a graph. This equality has consequences for both problems. On one hand, it allows us to prove the NP-completeness of the Cleaning problem, which was conjectured by Messinger et al. [M.-E. Messinger, R.J. Nowakowski, P. Pralat, Cleaning a network with brushes, Theoret. Comput. Sci. 399 (2008) 191-205]. On the other hand, it also enables us to design a faster algorithm for the Balanced Vertex-Ordering problem [J. Kara, K. Kratochvil, D. Wood, On the complexity of the balanced vertex ordering problem, Discrete Math. Theor. Comput. Sci. 9 (1) (2007) 193-202].

Parallel cleaning of a network with brushes

We consider the process of cleaning a network where at each time step, all vertices that have at least as many brushes as incident, contaminated edges, send brushes down these edges and remove them from the network. An added condition is that, because of the contamination model used, the final configuration must be the initial configuration of another cleaning of the network. We find the minimum number of brushes required for trees, cycles, complete bipartite networks; and for all networks when all edges must be cleaned on each step. Finally, we give bounds on the number of brushes required for complete networks.

Cleaning random d-regular graphs with brushes using a degree-greedy algorithm

In the recently introduced model for cleaning a graph with brushes, we use a degree-greedy algorithm to clean a random d-regular graph on n vertices (with dn even). We then use a differential equations method to find the (asymptotic) number of brushes needed to clean a random d-regular graph using this algorithm. As well as the case for general d, interesting results for specific values of d are examined. We also state various open problems.

Cleaning with Brooms

A model for cleaning a graph with brushes was recently introduced. Most of the existing papers consider the minimum number of brushes needed to clean a given graph G in this model, the so-called brush number b(G). In this paper, we focus on the broom number, B(G), that is, the maximum number of brushes that can be used to clean a graph G in this model.

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.

Fighting constrained fires in graphs

The firefighter problem is a simplified model for the spread of a fire (or disease or computer virus) in a network. A fire breaks out at a vertex in a connected graph, and spreads to each of its unprotected neighbours over discrete time-steps. A firefighter protects one vertex in each round which is not yet burned. While maximizing the number of saved vertices usually requires a strategy on the part of the firefighter, the fire itself spreads without any strategy. We consider a variant of the problem where the fire is constrained by spreading to a fixed number of vertices in each round. In the two-player game of k-firefighter, for a fixed positive integer k, the fire chooses to burn at most k unprotected neighbours in a given round. The k-surviving rate of a graph G is defined as the expected percentage of vertices that can be saved in k-firefighter when a fire breaks out at a random vertex of G. We supply bounds on the k-surviving rate, and determine its value for families of graphs including wheels and prisms. We show using spectral techniques that random d regular graphs have k-surviving rate at most (1+O(d^-^1^/^2))k+1. We consider the limiting surviving rate for countably infinite graphs. In particular, we show that the limiting surviving rate of the infinite random graph can be any real number in [1/(k+1),1].

The Robot Cleans Up

Imagine a large building with many corridors. A robot cleans these corridors in a greedy fashion, the next corridor cleaned is always the dirtiest to which it is adjacent. We determine bounds on the minimum s(G) and maximum S(G) number of time steps (over all edge weightings) before every edge of a graph G has been cleaned. We show that Eulerian graphs have a self-stabilizing property that holds for any initial edge weighting: after the initial cleaning of all edges, all subsequent cleanings require s(G) time steps. Finally, we show the only self-stabilizing trees are a subset of the superstars.

Elimination schemes and lattices

Abstract Perfect vertex elimination schemes are part of the characterizations for several classes of graphs, including chordal and cop-win. Partial elimination schemes reduce a graph to an important subgraph, for example, k -cores and robber-win graphs. We are interested in those partial elimination schemes, in which once a vertex is ready to be eliminated, it stays in that state regardless of which other vertices are eliminated. We show that in such a scheme, the sets of subsets of eliminated vertices, when ordered by inclusion, form an upper locally distributed lattice. We also show that (a) unless they contain a specific induced subgraph, the cop-win orderings have this property, and that (b) the process of cleaning graphs also leads to upper locally distributed lattices. Finally, we ask for an elimination scheme, which graphs are associated with distributive lattices?

A note on the eternal dominating set problem

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.

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.