The edge reconstruction number of a graph G, RN(G), is the minimum number of edge deleted subgraphs required to determine G up to isomorphism. We prove the following results for a disconnected graph G with at least two nontrivial components. If G has a pair of nontrivial nonisomorphic components then RN (G) ≤ 3. If G has a pair of nontrivial nonisomorphic components, is not a forest, and contai...

LetG = (V,E) be a simple undirected graph. We call any subset C ⊆ V an identifying code if the sets I(v) = {c ∈ C | {v, c} ∈ E or v = c} are distinct and non-empty for all vertices v ∈ V . A graph is called twin-free if there is an identifying code in the graph. The identifying code with minimum size in a twin-free graph G is called the optimal identifying code and the size of such a code is de...

A graph G with convex-QP stability number (or simply a convex-QP graph) is a graph for which the stability number is equal to the optimal value of a convex quadratic program, say P (G). There are polynomial-time procedures to recognize convex-QP graphs, except when the graph G is adverse or contains an adverse subgraph (that is, a non complete graph, without isolated vertices, such that the lea...

For the notion of neighbor-connectivity in graphs whenever a vertex is subverted the entire closed neighborhood of the vertex is deleted from the graph. The minimum number of vertices whose subversion results in an empty, complete, or disconnected subgraph is called the neighborconnectivity of the graph. Gunther, Hartnell, and Nowakowski have shown that for any graph, neighbor-connectivity is b...

The inference rule semi-resolution is introduced and shown to be complete for ground CNF formulas. It can be employed in a connection-graph setting, and preservation of the spanning property is shown to hold when activated links are deleted. Potential advantages from the employment of semi-resolution as the inference engine in prime implicate computations are brieey discussed .

We show that the planarity of a graph can be recognized from its vertex deleted subgraphs, which answers a question posed by Bondy and Hemminger in 1979. We also state some useful counting lemmas and use them to reconstruct certain planar graphs. AMS subject classification: Primary 05C60, Secondary 05C10, 05C40, 05C38

A graph G is called k-critical if it has chromatic number k; but every proper subgraph of G is ðk 1Þ-colourable. We prove that every k-critical graph ðkX6Þ on nXk þ 2 vertices has at least 1 2 ðk 1þ k 3 ðk cÞðk 1Þþk 3Þn edges where c 1⁄4 ðk 5Þð2 1 ðk 1Þðk 2ÞÞ: This improves earlier bounds established by Gallai (Acad. Sci. 8 (1963) 165) and by Krivelevich (Combinatorica 17 (1999) 401). r 2002 El...

The problem is to determine the number of ‘cops’ needed to capture a ‘robber’ where the game is played with perfect information, the different sides moving alternately. The cops capture the robber when one of them occupies the same vertex as the robber at any time in the game. A copwin graph is one in which one cop can always capture the robber. A graph is cop-win edge-critical with respect to ...