Fred R. McMorris

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 mathematical foundation for the analysis of cladistic character compatibility

Using formal algebraic definitions of “cladistic character” and “character compatibility”, the concept of “binary factors of a cladistic character” is formalized and used to describe and justify an algorithm for checking the compatibility of a set of characters. The algorithm lends itself to the selection of maximal compatible subsets when compatibility fails.

On probe interval graphs

Abstract Probe interval graphs have been introduced in the physical mapping and sequencing of DNA as a generalization of interval graphs. We prove that probe interval graphs are weakly triangulated, and hence are perfect, and characterize probe interval graphs by consecutive orders of their intrinsic cliques.

Triangulating Vertex-Colored Graphs

This paper examines the class of vertex-colored graphs that can be triangulated without the introduction of edges between vertices of the same color. This is related to a fundamental and long-standing problem for numerical taxonomists, called the Perfect Phylogeny Problem. These problems are known to be polynomially equivalent and NP-complete. This paper presents a dynamic programming algorithm that can be used to determine whether a given vertex-colored graph can be so triangulated and that runs in

The median procedure on median graphs

O((n+m(k-2))^{k+1})

An algebraic analysis of cladistic characters

time, where the graph has

Consensus functions defined on trees

n

When are two qualitative taxonomic characters compatible

vertices,

A View of Some Consensus Methods for Trees

m

When is one estimate of evolutionary relationships a refinement of another

edges, and