A complete graph is the graph in which every two vertices are adjacent. For a graph G = (V,E), the complete width of G is the minimum k such that there exist k independent sets Ni ⊆ V , 1 ≤ i ≤ k, such that the graph G obtained from G by adding some new edges between certain vertices inside the sets Ni, 1 ≤ i ≤ k, is a complete graph. The complete width problem is to decide whether the complete...

In this paper, we study the problem of designing in-place algorithms for finding the maximum clique in the intersection graphs of axis-parallel rectangles and disks in R2. First, we propose an O(n2 log n) time in-place algorithm for finding the maximum clique of the intersection graph of a set of n axis-parallel rectangles of arbitrary sizes. For the intersection graph of fixed height rectangle...

Let T = (V(T), E(T)) be a tree and B = {Pi} denote a collection of non-trivial (i.e., of length at least 1) simple paths in T, where a path P= (v,, u2, . . . , Q) is considered in the sequel as the collection {{uJ, {Q}, . . . ,{z)J, {u,, u2}, {u2, I+}, _ . . , {w,-,, Q}}. The intersection graph 0(S, 9) of 9 over set S has vertices which correspond to the paths in 9 and two vertices uk and U! ar...

A uniquely restricted matching is defined to be a matching M whose matched vertices induces a sub-graph which has only one perfect matching. In this paper, we make progress on the open question of the status of this problem on interval graphs (graphs obtained as the intersection graph of intervals on a line). We give an algorithm to compute maximum cardinality uniquely restricted matchings on c...

A disk graph is the intersection graph of disks in the plane, and a unit disk graph is the intersection graph of unit radius disks in the plane. We give upper and lower bounds on the number of labelled unit disk and disk graphs on n vertices. We show that the number of unit disk graphs on n vertices is n · α(n) and the number of disk graphs on n vertices is n · β(n), where α(n) and β(n) are Θ(1...

Given a system S of simple continuous curves (``strings'') in the plane, we can define a graph GS as follows. Assign a vertex to each curve, and connect two vertices by an edge if and only if the corresponding two curves intersect. GS is called the intersection graph of S. Not every graph is an intersection graph of a system of curves [EET76] (see Fig. 1 for a simple example). This implies that...

Abstract In this article we introduce a simple tool to derive polynomial upper bounds for the probability of observing unusually large maximal components in some models random graphs when considered at criticality. Specifically, apply our method model intersection graph, graph obtained through p -bond percolation on general d -regular and an inhomogeneous graph.

This work has two aims: First, we introduce a powerful technique for proving clique divergence when the graph satisfies a certain symmetry condition. Second, we prove that each closed surface admits a clique divergent triangulation. By definition, a graph is clique divergent if the orders of its iterated clique graphs tend to infinity, and the clique graph of a graph is the intersection graph o...

We first present new structural properties of a two-pair in various graphs. A twopair is used in a well-known characterization of weakly chordal graphs. Based on these properties, we prove the main theorem: a graph G is a weakly chordal (K2,3, 4P2, P2 ∪ P4, P6,H1,H2,H3)-free graph if and only if G is an edge intersection graph of subtrees on a tree with maximum degree 4. This characterizes the ...

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 i...