induced subgraphs
Author:
Jimmy Cao
Last Updated:
9 anni fa
License:
Creative Commons CC BY 4.0
Abstract:
induced subgraphs question
\begin
Discover why over 20 million people worldwide trust Overleaf with their work.
induced subgraphs question
\begin
Discover why over 20 million people worldwide trust Overleaf with their work.
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\begin{document}
For every connected undirected graph $G$, there exists a function $s_G$ with two parameters $n$ and $m$ defined like so:
\bigskip
$s_G(n, m)$ = Maximum number of distinct, connected subgraphs of $G$ of order $n$, in which each vertex of $G$ is used in at most $m$ of these subgraphs.
For example, in this graph:
\includegraphics[width=0.5\textwidth]{Download__2_.png}
$s(1, 1) = |V| = 5$
$s(2, \infty) = |E| = 4$
$s(2, 1) = 1$ (in fact for all diameter-2 graphs, $s(2, m) = m$ up to $|E|$)
Another useful property:
$s(|V| - 1, \infty) = $ number of non-articulation points in the graph.
\bigskip
My question is, does this $s_G$ function unique determine graph $G$?
In other words, are two graphs $G$ and $G'$ isomorphic if and only if they have the same function?
And if so, if you restrict the second parameter $m$ to the two values of $\{1, \infty\}$ does this new function also do the same?
\end{document}