Joshua D. Laison

Bar k-Visibility Graphs

Let S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a one-to-one correspondence between vertices of G and bars in S, such that there is an edge between two vertices in G if and only if there exists an unobstructed vertical line of sight between their corresponding bars. If bars are allowed to see through each other, the graphs representable in this way are precisely the interval graphs. We consider representations in which bars are allowed to see through at most k other bars. Since all bar visibility graphs are planar, we seek measurements of closeness to planarity for bar k-visibility graphs. We obtain an upper bound on the number of edges in a bar k-visibility graph. As a consequence, we obtain an upper bound of 12 on the chromatic number of bar 1-visibility graphs, and a tight upper bound of 8 on the size of the largest complete bar 1-visibility graph. We also consider the thickness of bar k-visibility graphs, obtaining an upper bound of 4 when k = 1, and a bound that is quadratic in k for k > 1.

Obstacle Numbers of Graphs

An obstacle representation of a graph G is a drawing of G in the plane with straight-line edges, together with a set of polygons (respectively, convex polygons) called obstacles, such that an edge exists in G if and only if it does not intersect an obstacle. The obstacle number (convex obstacle number) of G is the smallest number of obstacles (convex obstacles) in any obstacle representation of G. In this paper, we identify families of graphs with obstacle number 1 and construct graphs with arbitrarily large obstacle number (convex obstacle number). We prove that a graph has an obstacle representation with a single convex k-gon if and only if it is a circular arc graph with clique covering number at most k in which no two arcs cover the host circle. We also prove independently that a graph has an obstacle representation with a single segment obstacle if and only if it is the complement of an interval bigraph.

Bar k -visibility graphs: bounds on the number of edges, chromatic number, and thickness: bounds on the number of edges, chromatic number, and thickness

Let S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a one-to-one correspondence between vertices of G and bars in S, such that there is an edge between two vertices in G if and only if there exists an unobstructed vertical line of sight between their corresponding bars. If bars are allowed to see through each other, the graphs representable in this way are precisely the interval graphs. We consider representations in which bars are allowed to see through at most k other bars. Since all bar visibility graphs are planar, we seek measurements of closeness to planarity for bar k-visibility graphs. We obtain an upper bound on the number of edges in a bar k-visibility graph. As a consequence, we obtain an upper bound of 12 on the chromatic number of bar 1-visibility graphs, and a tight upper bound of 8 on the size of the largest complete bar 1-visibility graph. We conjecture that bar 1-visibility graphs have thickness at most 2.

Triangle, Parallelogram, and Trapezoid Orders

In his 1998 paper, Ryan classified the sets of unit, proper, and plain trapezoid and parallelogram orders. We extend this classification to include unit, proper, and plain triangle orders. We prove that there are 20 combinations of these properties that give rise to distinct classes of ordered sets, and order these classes by containment.

Finite prime distance graphs and 2-odd graphs

A graph G is a prime distance graph (respectively, a 2-odd graph) if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the dierence of their labels is prime (either 2 or odd). We prove that trees, cycles, and bipartite graphs are prime distance graphs, and that Dutch windmill graphs and paper mill graphs are prime distance graphs if and only if the Twin Prime Conjecture and dePolignac’s Conjecture are true, respectively. We give a characterization of 2-odd graphs in terms of edge colorings, and we use this characterization to determine which circulant graphs of the form Circ(n;f1;kg) are 2-odd and to prove results on circulant prime distance graphs.

Tube Representations of Ordered Sets

Abstract We define the (n,i,f)-tube orders, which include interval orders, trapezoid orders, triangle orders, weak orders, order dimension n, and interval-order-dimension n as special cases. We investigate some basic properties of (n,i,f)-tube orders, and begin classifying them by containment.

Prime power and prime product distance graphs

A graph

Unit and Proper Tube Orders

G

Fixing Numbers of Graphs and Groups

is a

Subspace intersection graphs

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