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We consider a deterministic discrete-time model of fire spread introduced by Hart-nell [1995] and the problem of minimizing the number of burnt vertices when deploying a limited number of firefighters per timestep. While only two firefighters per timestep are needed in the two dimensional lattice to contain any outbreak, we prove a conjecture of Wang and(More)
A bar visibility representation of a graph G is a collection of horizontal bars in the plane corresponding to the vertices of G such that two vertices are adjacent if and only if the corresponding bars can be joined by an unobstructed vertical line segment. In a bar k-visibility graph, two vertices are adjacent if and only if the corresponding bars can be(More)
Given a graph G, its triangular line graph is the graph T (G) with vertex set consisting of the edges of G and adjacencies between edges that are incident in G as well as being within a common triangle. Graphs with a representation as the triangular line graph of some graph G are triangular line graphs, which have been studied under many names including(More)
We define a natural class of graphs by generalizing prior notions of visibility, allowing the representing regions and sightlines to be arbitrary. We consider mainly the case of compact connected representing regions, proving two results giving necessary properties of visibility graphs, and giving some examples of classes of graphs that can be so(More)
Given a set F of graphs, a graph G is F-free if G does not contain any member of F as an induced subgraph. We say that F is a degree-sequence-forcing set if, for each graph G in the class C of F-free graphs, every realization of the degree sequence of G is also in C. We give a complete characterization of the degree-sequence-forcing sets F when F has(More)
An n-tuple π (not necessarily monotone) is graphic if there is a simple graph G with vertex set {v 1 ,. .. , v n } in which the degree of v i is the ith entry of π. Graphic n-tuples (d (2) n) pack if there are edge-disjoint n-vertex graphs G 1 and G 2 such that d G 1 (v i) = d (1) i and d G 2 (v i) = d (2) i for all i. We prove that graphic n-tuples π 1 and(More)
The distinguishing chromatic number χ D (G) of a graph G is the least integer k such that there is a proper k-coloring of G which is not preserved by any nontrivial automorphism of G. We study the distinguishing chromatic number of Cartesian products of graphs by focusing on how much it can exceed the trivial lower bound of the chromatic number χ(·). Our(More)
In a list of positive integers, let r and s denote the largest and smallest entries. A list is gap-free if each integer between r and s is present. We prove that a gap-free even-summed list is graphic if it has at least r + r+s+1 2s terms. With no restriction on gaps, length at least (r+s+1) 2 4s suffices, as proved by Zverovich and Zverovich. Both bounds(More)