Abstract

It is shown that the partition of a graph can be determined from its star polynomial and an algorithm is given for doing so. It is subsequently shown (as it is well known) that the partition of a graph is reconstructible from the set of node‐deleted subgraphs.

Highlights

  • The graphs here will be finite, undirected, and will have no loops or multiple m- edges

  • Let us associate with each m-star in G, an indeterminate or wig Wm+l; and wth each star cover

  • E(G;) Zw(C), where the summation is taken over all the star covers in G and

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Summary

INTRODUCTION

The graphs here will be finite, undirected, and will have no loops or multiple m- edges. E(G;) Zw(C), where the summation is taken over all the star covers in G and HG We will show that the partition of a graph G can be obtained from E(G;). This will be used to show that RG is node-reconstructible, a result that can be established by more elementary means (see Tutte [2]). We state a lemma which can be proved. Let v be a node of valency d in G () G contains m-stars with centre v. DEFINITION Let G be a graph with p nodes.

Let H G b
It follows that
Hence we get
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