William J. Lenhart

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A grid graph is a nite node-induced subgraph of the innnite two-dimensional integer grid. A solid grid graph is a grid graph without holes. For general grid graphs, the Hamiltonian cycle problem is known to be NP-complete. We give a polynomial-time algorithm for the Hamiltonian cycle problem in solid grid graphs, resolving a longstanding open question posed(More)
The link center of a simple polygon P is the set of points x inside P at which the maximal link-distance from x to any other point in P is minimized, where the link distance between two points x, y inside P is defined as the smallest number of straight edges in a polygonal path inside P connecting x to y. We prove several geometric properties of the link(More)
This paper examines an innnite family of proximity drawings of graphs called open and closed-drawings, rst deened by Kirkpatrick and Radke 15, 21] in the context of computational morphology. Such proximity drawings include as special cases the well-known Gabriel, relative neighborhood and strip drawings. Complete characterizations of those trees that admit(More)
Complete characterizations are given for those trees that can be drawn as either the relative neighborhood graph, relatively closest graph, Gabriel graph, or modified Gabriel graph of a set of points in the plane. The characterizations give rise to linear-time algorithms for determining whether a tree has such a drawing; if such a drawing exists one can be(More)
We show that there are, up to a trivial equivalence, precisely six theorems of the following form: If the vertices of a graph G are coloured red and white in such a way that no chordless path with four vertices is coloured in certain ways (specified by the particular theorem), then G is perfect if and only if each of the two subgraphs of G induced by all(More)