Terry A. McKee

Topics in intersection graph theory

Preface 1. Intersection Graphs. Basic Concepts Intersection Classes Parsimonious Set Representations Clique Graphs Line Graphs Hypergraphs 2. Chordal Graphs. Chordal Graphs as Intersection Graphs Other Characterizations Tree Hypergraphs Some Applications of Chordal Graphs Split Graphs 3. Interval Graphs. Definitions and Characterizations Interval Hypergraphs Proper Interval Graphs Some Applications of Interval Graphs 4. Competition Graphs. Neighborhood Graphs Competition Graphs Interval Competition Graphs Upper Bound Graphs 5. Threshold Graphs. Definitions and Characterizations Threshold Graphs as Intersection Graphs Difference Graphs and Ferrers Digraphs Some Applications of Threshold Graphs 6. Other Kinds of Intersection. p-Intersection Graphs Intersection Multigraphs and Pseudographs Tolerance Intersection Graphs 7. Guide to Related Topics. Assorted Geometric Intersection Graphs Bipartite Intersection Graphs, Intersection Digraphs, and Catch (Di)Graphs Chordal Bipartite and Weakly Chordal Graphs Circle Graphs and Permutation Graphs Clique Graphs of Chordal Graphs and Clique-Helly Graphs Containment, Comparability, Cocomparability, and Asteroidal Triple-Free Graphs Infinite Intersection Graphs Miscellaneous Topics P4-Free Chordal Graphs and Cographs Powers of Intersection Graphs Sphere-of-Influence Graphs Strongly Chordal Graphs Bibliography Index.

A fast method dispatcher for compiled languages with multiple inheritance

This paper addresses the problem of an efficient dispatch mechanism in an object-oriented system with multiple inheritance. The solution suggested is a direct table indexed branch such as is used in C++. The table slot assignments are made using a coloring algorithm. The method is applicable to strongly typed languages such as C++ (with multiple inheritance added) and Eiffel, and in a slightly slower form to less strongly typed languages like Objective C.

The leafage of a chordal graph

The leafage l(G) of a chordal graph G is the minimum number of leaves of a tree in which G has an intersection representation by subtrees. We obtain upper and lower bounds on l(G) and compute it on special classes. The maximum of l(G) on n-vertex graphs is n − lg n− 1 2 lg lg n+O(1). The proper leafage l(G) is the minimum number of leaves when no subtree may contain another; we obtain upper and lower bounds on l(G). Leafage equals proper leafage on claw-free chordal graphs. We use asteroidal sets and structural properties of chordal graphs.

Recharacterizing Eulerian: Intimations of new duality

Abstract Eulerian graphs are shown to be characterized by being connected with each edge in an odd number of circuits, as compared with the traditional characterization having each cutset contain an even number of edges. This result is proved in the general context of binary matroids, and the intriguing sort of duality present is analyzed using syntactical duality principles.

p-competition graphs

Abstract If D = ( V , A ) is a digraph, its p -competition graph has vertex set V and an edge between x and y if and only if there are distinct vertices a 1 , …, a p and arcs ( x , a i ) and ( y , a i ) in D for each i ≤ p . This definition generalizes the widely studied p = 1 case of ordinary competition graphs. We obtain results about p -competition graphs analogous to the well-known results about ordinary competition graphs and apply these results to specific cases.

Representations of graphs modulo n

A graph is representable modulo n if its vertices can be labeled with distinct integers between 0 and n, the difference of the labels of two vertices being relatively prime to n if and only if the vertices are adjacent. Erdős and Evans recently proved that every graph is representable modulo some positive integer. We derive a combinatorial formulation of representability modulo n and use it to characterize those graphs representable modulo certain types of integers, in particular integers with only two prime divisors. Other facets of representability are also explored. We obtain information about the values of n modulo which paths and cycles are representable.

2-competition graphs

IfD (V, A) is a digraph, its p-competition graph for p a positive integer has vertex set V and an edge between x and y ifand only if there are distinct vertices a, , an in D with (x, a and (y, a) arcs ofD for each 1, , p. This notion generalizes the notion of ordinary competition graph, which has been widely studied and is the special case wherep 1. Results about the case wherep 2 are obtained. In particular, the paper addresses the question ofwhich complete bipartite graphs are 2-competition graphs. This problem is formulated as the following combinatorial problem: Given disjoint setsA and B such that A tO BI n, when can one find n subsets ofA tO B so that every a in A and b in B are together contained in at least two of the subsets and so that the intersection of every pair of subsets contains at most one element from A and at most one element from B?

Structural conditions for cycle completable graphs

Abstract We first review the matrix completion problem that motivated cycle completable graphs. We then discuss prior and give new structural characterizations of these graphs, along with their relationship to chordal and series-parallel graphs.

Edge-clique graphs

The edge-clique graphK(G) of a graphG is that graph whose vertices correspond to the edges ofG and where two vertices ofK(G) are adjacent whenever the corresponding edges ofG belong to a common clique. It is shown that every edge-clique graph is a clique graph, and that ifG is either an interval graph or a line graph, then so too isK(G). An algorithm is provided for determining whether a graph is an edge-clique graph. A new graph called the STP graph is introduced and a relationship involving this graph, the edge-clique graph, and the line graph is presented. The STP graphs are also characterized.

p -competition numbers

Abstract If D = ( V , A >) is a digraph, its p -competition graph has vertex set V and an edge between x and y if and only if there are distinct vertices a 1 ,…, a p in D with ( x, a i ) and ( y,a i ) arcs of D for each i ⩽ p . The p -competition number of a graph is the smallest number of isolated vertices which need to be added in order to make it a p -competition graph. These notions generalize the widely studied p = 1 case, where they correspond to ordinary competition graphs and competition numbers. We obtain bounds on the p -competition number in terms of the ordinary competition number, and show that, surprisingly, the p -competition number can be arbitrarily smaller than the ordinary competition number.