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The well-known Friendship Theorem states that if G is a graph in which every pair of vertices has exactly one common neighbor, then G has a single vertex joined to all others (a " universal friend "). V. Sós defined an analogous friendship property for 3-uniform hypergraphs, and gave a construction satisfying the friendship property that has a universal… (More)

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)

A (finite) sequence of nonnegative integers is graphic if it is the degree sequence of some simple graph G. Given graphs G 1 and G 2 , we consider the smallest integer n such that for every n-term graphic sequence π, there is some graph G with degree sequence π with G 1 ⊆ G or with G 2 ⊆ G. When the phrase " some graph " in the prior sentence is replaced… (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)

We consider three graph-theoretical models of spread and competition motivated by biological applications and also by spread of opinions in social networks and fires in geographical areas. We modify the voter model of Clifford, Sudbury, Holley, and Liggett by introducing confidence levels. In the voter model, a group of voters has opinions " yes " or " no,… (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)