A module of an undirected graph is a set X of nodes such for each node x not in X , either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linear-space representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a directi...

A transitive orientation of a graph is an orientation of the edges that produces a transitive digraph. The modular decomposition of a graph is a canonical representation of all of its modules. Finding a transitive orientation and finding the modular decomposition are in some sense dual problems. In this paper, we describe a simple O(n+mlogn) algorithm that uses this duality to find both a trans...

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v, u, x, y) of vertices such that both (v, u, x) and (u, x, y) are paths of length two. The 3-arc graph of a graph G is defined to have vertices the arcs of G such that two arcs uv, xy are adjacent if and only if (v, u, x, y) is a 3-arc of G. In this paper we prove that any connected 3-arc graph is Hamiltonian, and all iterative 3-...

Several questions and few answers regarding percolation on finite graphs are presented. The following is a note regarding the asymptotic study of percolation on finite transitive graphs. On the one hand, the theory of percolation on infinite graphs is rather developed, although still with many open problems (See [9]). On the other hand random graphs were deeply studied (see [7]). Finite transit...

‎let $g=(v(g),e(g))$ be a simple connected graph with vertex set $v(g)$ and edge‎ ‎set $e(g)$‎. ‎the (first) edge-hyper wiener index of the graph $g$ is defined as‎: ‎$$ww_{e}(g)=sum_{{f,g}subseteq e(g)}(d_{e}(f,g|g)+d_{e}^{2}(f,g|g))=frac{1}{2}sum_{fin e(g)}(d_{e}(f|g)+d^{2}_{e}(f|g)),$$‎ ‎where $d_{e}(f,g|g)$ denotes the distance between the edges $f=xy$ and $g=uv$ in $e(g)$ and $d_{e}(f|g)=s...

The geodesic and geodesic interval, namely the set of all vertices lying on geodesics between a pair of vertices in a connected graph, is a part of folklore in metric graph theory. It is also known that Steiner tree of a (multi) set with k (k > 2) vertices, generalizes geodesics. In [1] the authors studied the k-Steiner intervals S(u1, u2, . . . , uk) on connected graphs (k ≥ 3) as the k-ary ge...

This paper presents a novel approach to represent spatiotemporal visual information. We introduce a surface-based shape model whose structure is invariant to surface variations over time to describe 3D dynamic surfaces (e.g., obtained from multiview video capture). The descriptor is defined as a graph lying on object surfaces and anchored to invariant local features (e.g., extremal points). Geo...