Stephen G. Hartke

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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)
A hub set in a graph G is a set U ⊆ V (G) such that any two vertices outside U are connected by a path whose internal vertices lie in U . We prove that h(G) ≤ hc(G) ≤ γc(G) ≤ h(G) + 1, where h(G), hc(G), and γc(G), respectively, are the minimum sizes of a hub set in G, a hub set inducing a connected subgraph, and a connected dominating set. Furthermore, all(More)
We consider a deterministic discrete-time model of fire spread introduced by Hartnell [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)
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)
An n-tuple π (not necessarily monotone) is graphic if there is a simple graph G with vertex set {v1, . . . , vn} in which the degree of vi is the ith entry of π. Graphic n-tuples (d (1) 1 , . . . , d (1) n ) and (d (2) 1 , . . . , d (2) n ) pack if there are edge-disjoint n-vertex graphs G1 and G2 such that dG1(vi) = d (1) i and dG2(vi) = d (2) i for all i.(More)
A graph G is uniquely Kr-saturated if it contains no clique with r vertices and if for all edges e in the complement, G + e has a unique clique with r vertices. Previously, few examples of uniquely Kr-saturated graphs were known, and little was known about their properties. We search for these graphs by adapting orbital branching, a technique originally(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)
A sequence of nonnegative integers is k-graphic if it is the degree sequence of a kuniform hypergraph. The only known characterization of k-graphic sequences is due to Dewdney in 1975. As this characterization does not yield an efficient algorithm, it is a fundamental open question to determine a more practical characterization. While several necessary(More)