Generalized degeneracy, dynamic monopolies and maximum degenerate subgraphs

A graph $G$ is said to be a $k$-degenerate graph if any subgraph of $G$ contains a vertex of degree at most $k$. Let $κ$ be any non-negative function on the vertex set of $G$. We first define a $κ$-degenerate graph. Next we give an efficient algorithm to determine whether a graph is $κ$-degenerate. We revisit the concept of dynamic monopolies in graphs. The latter notion is used in formulation and analysis of spread of influence such as disease or opinion in social networks. We consider dynamic monopolies with (not necessarily positive) but integral threshold assignments. We obtain a sufficient and necessary relationship between dynamic monopolies and generalized degeneracy. As applications of the previous results we consider the problem of determining the maximum size of $κ$-degenerate (or $k$-degenerate) induced subgraphs in any graph. We obtain some upper and lower bounds for the maximum size of any $κ$-degenerate induced subgraph in general and regular graphs. All of our bounds are constructive.

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