A graph is an interval graph if it is the intersection graph of intervals on a line. Interval graphs are known to be the intersection of chordal graphs and asteroidal triple-free graphs, two families where the well-known Lexicographic Breadth First Search (LBFS) plays an important algorithmic and structural role. In this paper we show that interval graphs have a very rich LBFS structure and tha...

‎let $g$ be a finite group which is not a cyclic $p$-group‎, ‎$p$ a prime number‎. ‎we define an undirected simple graph $delta(g)$ whose‎ ‎vertices are the proper subgroups of $g$, which are not contained in the‎ ‎frattini subgroup of $g$ and two vertices $h$ and $k$ are joined by an edge‎ ‎if and only if $g=langle h‎ , ‎krangle$‎. ‎in this paper we classify finite groups with planar graph‎. ‎...

A caterpillar graph is a tree in which the removal of all pendant vertices results in a chordless path. In this work, we determine the number of maximal independent sets (mis) in caterpillar graphs. For a general graph, this problem is #P—complete. We provide a polynomial time algorithm to generate the whole family of mis in a caterpillar graph. We also characterize the independent graph (inter...

The perfect phylogeny problem is a classic problem in computational biology, where we seek an unrooted phylogeny that is compatible with a set of qualitative characters. Such a tree exists precisely when an intersection graph associated with the character set, called the partition intersection graph, can be triangulated using a restricted set of fill edges. Semple and Steel used the partition i...

For two xed graphs X and Y the X Y intersection graph of a graph G is a graph whose vertices are induced subgraphs of G isomor phic to Y and where two vertices are adjacent if their intersection in G contains an induced subgraph isomorphic to X A conformal k graph is a simple hypergraph whose hyperedges are exactly k cliques in the section of the hypergraph Let kX denote the disjoint union of k...

We give a characterization for fundamental groups of graphs of groups amalgamating cyclic edge subgroups to be cyclic subgroup separable if each pair of edge subgroups has a non-trivial intersection. We show that fundamental groups of graphs of abelian groups amalgamating cyclic edge subgroups are cyclic subgroup separable, hence residually finite, if each edge subgroup is isolated in its conta...

A cycle C of length k is extendable if there is a cycle C′ of length k+1 with V (C) ⊂ V (C′). A graph G = (V,E) of order n is cycle extendable when every cycle C of length k < n is extendable. A chordal graph is a spider intersection graph if it admits an intersection representation which consists of subtrees of a sub-divided star (or spider). In 1990, Hendry conjectured that all hamiltonian ch...

In this report, we present the intersection graphs and describe the role they play in computations of treewidth. We introduce minimal triangulations, minimal separators, potential maximal cliques and blocks as the main tools for exact computation and approximation of that important graph parameter. The first chapter is an introduction to the intersection graph theory. We give all the necessary ...

An intersection representation of a graph G is a function f : V…G† ! 2 (where S is any set) with the property that uv A E…G† if and only if f …u†V f …v†0h. The size of the representation is jSj. The intersection number of G is the smallest size of an intersection representation of G. The intersection number can be expressed as an integer program, and the value of the linear relaxation of that p...