We introduce graphs called hypochordal : for any path of length 2, there exists a chord or another path of length 2 between its two endpoints. We show that such graphs are 2-vertex-connected and moreover in the case of an edge or a vertex deletion, the distance between any pair of nonadjacent vertices remains unchanged. We give properties of hypochordal graphs, then we study the class of minimu...

An ecard of a graph G is a subgraph formed by deleting an edge. A da-ecard specifies the degree of the deleted edge along with the ecard. The degree associated edge reconstruction number of a graph G, dern(G), is the minimum number of da-ecards that uniquely determines G. The adversary degree associated edge reconstruction number of a graph G, adern(G), is the minimum number k such that every c...

The atom-bond connectivity index is a useful topological index in studying the stability of alkanes and the strain energy of cycloalkanes. In this paper some inequalities for the atom-bond connectivity index of a series of graph operations are presented. We also prove our bounds are tight. As an application, the ABC indices of C4 nanotubes and nanotori are computed. © 2011 Elsevier B.V. All rig...

Seidel’s switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other one by a sequence of switches. Jelínková et al. [DMTCS 13, no. 2, 2011] presented a proof that it is NP-complete to decide ...

This paper concerns the tau constant, which is an important invariant of a metrized graph, and which has applications to arithmetic properties of curves. We give several formulas for the tau constant, and show how it changes under graph operations including deletion of an edge, contraction of an edge, and union of graphs along one or two points. We show how the tau constant changes when edges o...

Lubiw [11] conjectures that in a minimal imperfect Berge graph, the neighborhood graph N (v) of any vertex v must be connected; this conjecture implies a well known Chvátal’s Conjecture [6] which states that N (v) must contain a P4. In this note we will prove an intermediary conjecture for some classes of minimal imperfect graphs. It is well known that a graph is P4-free if, and only if, every ...

A graph G is 3-e.c. if for each distinct triple S of vertices, and each subset T of S, there is a vertex not in S joined to the vertices of T and to no other vertices of S. Few explicit examples of 3-e.c. graphs are known, although almost all graphs are 3-e.c. We provide new examples of 3-e.c. graphs arising as incidence graphs of partial planes resulting from affine planes. We also present a n...

We examine graphs that, for every spatial embedding, have a pair of linked cycles such that linked cycles do not pass through every vertex of the graph (we call such a graph intrinsically linked with an unused vertex). We demonstrate three graph operations that transform an intrinsically linked graph to a graph with this property. It remains an open question to determine all minor-minimal graph...

Motivated by the problem of zero-error broadcasting, we introduce a new notion of graph capacity, termed ρ-capacity, that generalizes the Shannon capacity of a graph. We derive upper and lower bounds on the ρ-capacity of arbitrary graphs, and provide a Lovász-type upper bound for regular graphs. We study the behavior of the ρ-capacity under two graph operations: the strong product and the disjo...