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A Reduction of the Graph Reconstruction Conjecture

S. Monikandan, J. Balakumar (2014)

Discussiones Mathematicae Graph Theory

A graph is said to be reconstructible if it is determined up to isomor- phism from the collection of all its one-vertex deleted unlabeled subgraphs. Reconstruction Conjecture (RC) asserts that all graphs on at least three vertices are reconstructible. In this paper, we prove that interval-regular graphs and some new classes of graphs are reconstructible and show that RC is true if and only if all non-geodetic and non-interval-regular blocks G with diam(G) = 2 or diam(Ḡ) = diam(G) = 3 are reconstructible...

A remark on branch weights in countable trees

Bohdan Zelinka (2004)

Mathematica Bohemica

Let T be a tree, let u be its vertex. The branch weight b ( u ) of u is the maximum number of vertices of a branch of T at u . The set of vertices u of T in which b ( u ) attains its minimum is the branch weight centroid B ( T ) of T . For finite trees the present author proved that B ( T ) coincides with the median of T , therefore it consists of one vertex or of two adjacent vertices. In this paper we show that for infinite countable trees the situation is quite different.

A remark on graph operators

Bohdan Zelinka (1999)

Mathematica Bohemica

A theorem is proved which implies affirmative answers to the problems of E. Prisner. One problem is whether there are cycles of the line graph operator L with period other than 1, the other whether there are cycles of the 4-edge graph operator ∇ 4 with period greater than 2. Then a similar theorem follows.

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