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Cyclomatic numbers of connected induced subgraphs. (English) Zbl 0783.05065

For a graph \(G=\bigl( V(G),E(G) \bigr)\), the cyclomatic number \(cy(G)\) is defined by \(cy(G)=| E(G) |-| V(G) |+1\). Let \(A\) be an independent set of vertices of \(G\) and let \(C(A)\) be the collection of all connected induced subgraphs of \(G\) which contain \(A\). Define \(\omega(A)=\min \bigl\{ cy(H):H \in C(A) \bigr\}\). The author obtains upper bounds for \(\omega(A)\) over the class of graphs and over the class of triangle-free graphs. He also considers the edge version of the question and shows that all upper bounds are best possible.

MSC:

05C35 Extremal problems in graph theory
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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References:

[1] Alspach, B.; Oral, H., Research Problem 95, Discrete Math., 71, 185 (1988)
[2] Bondy, J. A.; Murty, U. S.R., Graph Theory with Applications (1976), Elsevier: Elsevier Amsterdam · Zbl 1134.05001
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