Abstract
Given a set ξ = {H 1,H 2, ⋯ } of connected non-acyclic graphs, a ξ-free graph is one which does not contain any member of ξ as induced subgraph. Our first purpose in this paper is to perform an investigation into the limiting distribution of labeled graphs and multigraphs (graphs with possible self-loops and multiple edges), with n vertices and approximately \(\frac{1}{2}n\) edges, in which all sparse connected components are ξ-free. Next, we prove that for any finite collection ξ of multicyclic graphs almost all connected graphs with n vertices and n + o(n 1/3) edges are ξ -free. The same result holds for multigraphs.
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Ravelomanana, V., Thimonier, L. (2000). Some Remarks on Sparsely Connected Isomorphism-Free Labeled Graphs. In: Gonnet, G.H., Viola, A. (eds) LATIN 2000: Theoretical Informatics. LATIN 2000. Lecture Notes in Computer Science, vol 1776. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10719839_3
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DOI: https://doi.org/10.1007/10719839_3
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