On the Reconstruction of Graphs

Discrete Mathematics, 2003

In this paper we de2ne 2ve families of simple graphs {fi,f 2 ,f^,t^ and F= 5) such that the reconstruction conjecture is true if reconstruction is proved for either the families $[,$2 m & % or the families F=3 ; F=4 and F= 5. These families are quite restrictive in that each has diameter two or three. Then we prove that families F = 1 ; F= 2; F=3 ; F= 4 and F= 5 are recognizable by proving that graphs of diameter two and edge-minimal graphs of diameter two are recognizable. The reconstruction conjecture is thus reduced to showing weak reconstructibility for either of the two sets of families.

Australasian Journal of Combinatorics, 1997

With the help of a novel computational technique, we show that graphs with up to 11 vertices are determined uniquely by their sets of vertex-deleted subgraphs, even if the set of subgraphs is reduced by isomorphism type. The same result holds for trianglefree graphs to 14 vertices, square-free graphs to 15 vertices and bipartite graphs to 15 vertices, as well as some other classes.

Discrete Mathematics, 1983

Electronic Journal of Combinatorics, 2000

The question of whether the characteristic polynomial of a simple graph is uniquely determined by the characteristic polynomials of its vertex-deleted subgraphs is one of the many unresolved problems in graph reconstruction. In this paper we prove that the characteristic polynomial of a graph is reconstructible from the characteristic polynomials of the vertex-deleted subgraphs of the graph and its complement.

A graph is said to be reconstructible if it is determined up to isomorphism from the collection of all its one-vertex deleted unlabeled subgraphs. The reconstruction conjecture (RC) asserts that all graphs on at least three vertices are reconstructible. In this paper, we prove that all triangle-free graphs G with connectivity two such that diam(G)=2 or diam(G)=diamG ¯=3 are reconstructible. We also prove that all 2-connected bipartite graphs G such that diam(G)=2 or diam(G)=diamG ¯=3 are reconstructible and show that RC is true if and only if all 2-connected graphs H containing an odd cycle such that diam(H)=2 or diam(H)=diamH ¯=3 are reconstructible.

Ars Combinatoria, 2002

The reconstruction number rn(G) of graph G is the minimum number of vertex-deleted subgraphs of G required in order to identify G up to isomporphism. Myrvold and Molina have shown that if G is disconnected and not all components are isomorphic then rn(G) = 3, whereas, if all components are isomorphic and have c vertices each, then rn(G) can be as large as c + 2. In this paper we propose and initiate the study of the gap between rn(G) = 3 and rn(G) = c + 2. Myrvold showed that if G consists of p copies of K c , then rn(G) = c + 2. We show that, in fact, this is the only class of disconnected graphs with this value of rn(G). We also show that if rn(G) c + 1 (where c is still the number of vertices in any component), then, again, G can only be copies of K ≥ c. It then follows that there exist no disconnected graphs G with c vertices in each component and rn(G) = c + 1. This poses the problem of obtaining for a given c, the largest value of t = t(c) such that there exists a disconnected graph with all components of order c, isomorphic and not equal to K c and is such that rn(G) = t.

It is proved that the set of branches of a graph G is reconstructible except in a very special case. More precisely the set of branches of a graph G is reconstructible unless all the following hold: (1) the pruned center of G is a vertex or an edge, (2) G has exactly two branches, (3) one branch contains all the vertices of degree one of G and the other branch contains exactly one end-block. This is the best possible result in the sense that in the special excluded case, the reconstruction of the set of branches is equivalent to the reconstruction of the graph itself.

Discrete Applied Mathematics, 2007

We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertex-deleted or edge-deleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related for all amounts c ≥

A sequence α of nonnegative integers can represent degrees of a graph G and β for the graph H. there may be many different 1-to-1 or 1-to-many mapping functions by which G can be mapped into H. That is it is feasible to construct isomorphic or regular or connected or disconnected graphs. Finding connectedness of a graph from degree sequence is analogues to Reconstruction Conjecture problem. It is our intention in this paper to infer about the connectedness of the graph only from the degree sequence and no need of any other information. It is evident that there is no unique conclusion about the connectedness of a given graph from the algorithm we project here. However, we can say that whether the sequence represents a connected or disconnected graph.

Discrete Mathematics, 2007

We examine decompositions of complete graphs K 4k+2 into 2k+1 isomorphic spanning trees. We develop a method of factorization based on a new type of vertex labelling, namely blended -labelling. We also show that for every k 1 and every d, 3 d 4k + 1 there is a tree with diameter d that decomposes K 4k+2 into 2k + 1 factors isomorphic to T .