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Factoring a graph in polynomial time. (English) Zbl 0625.05050
The Cartesian product \(G\times H\) of graphs G and H has vertices (g,h) where g is a vertex in G and h a vertex in H. Two vertices of \(G\times H\), say \((g_ 1,h_ 1)\) and \((g_ 2,h_ 2)\), are connected by an edge in \(G\times H\), just when either \(\{g_ 1,g_ 2\}\) is an edge of G and \(h_ 1=h_ 2\), or when \(g_ 1=g_ 2\) and \(\{h_ 1,h_ 2\}\) is an edge of H. It was proved by G. Sabidussi [Math. Z. 72, 446-457 (1960; Zbl 0093.376)] that Cartesian product admits unique factorization. The author proves that there is a polynomial time algorithm for constructing the prime factorization of a given connected graph. The same result, using completely different techniques, was proved also in [J. Feigenbaum, J. Hershberger and A. A. Schaffer, Discrete Appl. Math. 12, 123-138 (1985; Zbl 0579.68028)].
Reviewer: G.Slutzki

MSC:
05C99 Graph theory
68R10 Graph theory (including graph drawing) in computer science
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References:
[1] J. Feigenbaum, J. Hershberger, A. Schaffer, A polynomial time algorithm for finding the prime factors of Cartesian-product graphs, Disc. Appl. Math. 12 (2), 123-138. · Zbl 0579.68028
[2] R.L. Graham, P.M. Winkler, On isometric embeddings of graphs, Trans. Amer. Math. Soc. 288 (2), 527-536. · Zbl 0576.05017
[3] Imrich, W., ()
[4] Sabidussi, G., Graph multiplication, Math. zeitschr., 72, 446-457, (1960) · Zbl 0093.37603
[5] Welsh, D., Problems in computational complexity, (), 75-85
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