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.Comment: 19 pages, 3 figure

Fixed cardinality stable sets

Given an undirected graph G=(V,E) and a positive integer k in {1, ..., |V|}, we initiate the combinatorial study of stable sets of cardinality exactly k in G. Our aim is to instigate the polyhedral investigation of the convex hull of fixed cardinality stable sets, inspired by the rich theory on the classical structure of stable sets. We introduce a large class of valid inequalities to the natural integer programming formulation of the problem. We also present simple combinatorial relaxations based on computing maximum weighted matchings, which yield dual bounds towards finding minimum-weight fixed cardinality stable sets, and particular cases which are solvable in polynomial time.publishedVersio

Simplicial decompositions of graphs: a survey of applications

AbstractWe survey applications of simplicial decompositions (decompositions by separating complete subgraphs) to problems in graph theory. Among the areas of application are excluded minor theorems, extremal graph theorems, chordal and interval graphs, infinite graph theory and algorithmic aspects

Hamiltonicity, Pancyclicity, and Cycle Extendability in Graphs

The study of cycles, particularly Hamiltonian cycles, is very important in many applications. Bondy posited his famous metaconjecture, that every condition sufficient for Hamiltonicity actually guarantees a graph is pancyclic. Pancyclicity is a stronger structural property than Hamiltonicity. An even stronger structural property is for a graph to be cycle extendable. Hendry conjectured that any graph which is Hamiltonian and chordal is cycle extendable. In this dissertation, cycle extendability is investigated and generalized. It is proved that chordal 2-connected K1,3-free graphs are cycle extendable. S-cycle extendability was defined by Beasley and Brown, where S is any set of positive integers. A conjecture is presented that Hamiltonian chordal graphs are {1, 2}-cycle extendable. Dirac’s Theorem is an classic result establishing a minimum degree condition for a graph to be Hamiltonian. Ore’s condition is another early result giving a sufficient condition for Hamiltonicity. In this dissertation, generalizations of Dirac’s and Ore’s Theorems are presented. The Chvatal-Erdos condition is a result showing that if the maximum size of an independent set in a graph G is less than or equal to the minimum number of vertices whose deletion increases the number of components of G, then G is Hamiltonian. It is proved here that the Chvatal-Erdos condition guarantees that a graph is cycle extendable. It is also shown that a graph having a Hamiltonian elimination ordering is cycle extendable. The existence of Hamiltonian cycles which avoid sets of edges of a certain size and certain subgraphs is a new topic recently investigated by Harlan, et al., which clearly has applications to scheduling and communication networks among other things. The theory is extended here to bipartite graphs. Specifically, the conditions for the existence of a Hamiltonian cycle that avoids edges, or some subgraph of a certain size, are determined for the bipartite case. Briefly, this dissertation contributes to the state of the art of Hamiltonian cycles, cycle extendability and edge and graph avoiding Hamiltonian cycles, which is an important area of graph theory

固有値分解とテンソル分解を用いた大規模グラフデータ分析に関する研究

筑波大学 (University of Tsukuba)201