Robert E. Jamison

Convexity in graphs and hypergraphs

We study several notions of abstract convexity in graphs and hypergraphs. In each case, we obtain analogues of several classical results, including the Minkowski–Krein–Milman theorem, Caratheodory’s theorem and Tietze’s convexity theorem. In addition, our results yield new characterizations of the classes of chordal gaphs, strongly chordal graphs, Ptolemaic graphs and totally balanced hypergraphs.

The edge intersection graphs of paths in a tree

Abstract The class of edge intersection graphs of a collection of paths in a tree (EPT graphs) is investigated, where two paths edge intersect if they share an edge. The cliques of an EPT graph are characterized and shown to have strong Helly number 4. From this it is demonstrated that the problem of finding a maximum clique of an EPT graph can be solved in polynomial time. It is shown that the strong perfect graph conjecture holds for EPT graphs. Further complexity results follow from the observation that every line graph is an EPT graph. The class of EPT graphs is equivalent to the class of fundamental cycle graphs.

On local convexity in graphs

Abstract A set K of nodes of a graph G is geodesically convex (respectively, monophonically convex) if K contains every node on every shortest (respectively, chordless) path joining nodes in K. We investigate the classes of graphs which are characterized by certain local convexity conditions with respect to geodesic convexity, in particular, those graphs in which balls around nodes are convex, and those graphs in which neighborhoods of convex sets are convex. For monophonic convexity, these conditions are known to be equivalent, and hold if and only if the graph is chordal. Although these conditions are not equivalent for geodesic convexity, each defines a generalization of the class of chordal graphs. A persistent theme here will be the analogies between these graphs and chordal graphs.

Edge and vertex intersection of paths in a tree

Abstract In this paper we continue the investigation of the class of edge intersection graphs of a collection of paths in a tree (EPT graphs) where two paths edge intersect if they share an edge. The class of EPT graphs differs from the class known as path graphs, the latter being the class of vertex intersection graphs of paths in a tree. A characterization is presented here showing when a path graph is an EPT graph. In particular, the classes of path graphs and EPT graphs coincide on trees all of whose vertices have degree at most 3. We then prove that it is an NP-complete problem to recognize whether a graph is an EPT graph.

The subchromatic number of a graph

The subchromatic number X S ( G ) of a graph G = ( V, E ) is the smallest order k of a partition {V 1 , V 2 , …, V k } of the vertices V ( G ) such that the subgraph ( V i ) induced by each subset V i consists of a disjoint union of complete subgraphs. By definition, X s ( G ) ⩽ ( G ), the chromatic number of G . This paper develops properties of this lower bound for the chromatic number.

On the average number of nodes in a subtree of a tree

Abstract For any tree T (labelled, not rooted) of order n , it will be shown that the average number of nodes in a subtree of T is at least (n + 2) 3 , with this minimum achieved iff T is a path. For T rooted, the average number of nodes in a subtree containing the root is at least (n + 1) 2 and always exceeds the average over all unrooted subtrees. For the maximum mean, examples show that there are arbitrarily large trees in which the average subtree contains essentially all of the nodes. The mean subtree order as a function on trees is also shown to be monotone with respect to inclusion.

Tolerance intersection graphs on binary trees with constant tolerance

Abstract A chordal graph is the intersection graph of a family of subtrees of a tree, or, equivalently, it is the (edge-)intersection graph of leaf-generated subtrees of a full binary tree. In this paper, a generalization of chordal graphs from this viewpoint is studied: a graph G=(V,E) is representable if there is a family of subtrees {Sv}v∈V of a binary tree, such that uv∈E if and only if |Su∩Sv|⩾3.

Constant tolerance intersection graphs of subtrees of a tree

Abstract A chordal graph is the intersection graph of a family of subtrees of a host tree. In this paper we generalize this. A graph G = ( V , E ) has an ( h , s , t ) -representation if there exists a host tree T of maximum degree at most h , and a family of subtrees { S v } v ∈ V of T , all of maximum degree at most s , such that uv ∈ E if and only if | S u ∩ S v | ⩾ t . For given h , s , and t , there exist infinitely many forbidden induced subgraphs for the class of ( h , s , t ) -graphs. On the other hand, for fixed h ⩾ s ⩾ 3 , every graph is an ( h , s , t ) -graph provided that we take t large enough. Under certain conditions representations of larger graphs can be obtained from those of smaller ones by amalgamation procedures. Other representability and non-representability results are presented as well.

A Helly theorem for convexity in graphs

Abstract It is shown that for chordless path convexity in any graph, the Helly number equals the size of a maximum clique.

Closure spaces that are not uniquely generated

Because antimatroid closure spaces satisfy the anti-exchange axiom, it is easy to show that they are uniquely generated. That is, the minimal set of elements determining a closed set is unique. A prime example is a discrete convex geometry in Euclidean space where closed sets are uniquely generated by their extreme points. But, many of the geometries arising in computer science, e.g. the world wide web or rectilinear VLSI layouts are not uniquely generated. Nevertheless, these closure spaces still illustrate a number of fundamental antimatroid properties which we demonstrate in this paper. In particular, we examine both a pseudo-convexity operator and the Galois closure of formal concept analysis. In the latter case, we show how these principles can be used to automatically convert a formal concept lattice into a system of implications.