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 $\kappa$ be any non-negative function on the vertex set of $G$. We first define a $\kappa$-degenerate graph. Next we give an efficient algorithm to determine whether a graph is $\kappa$-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 $\kappa$-degenerate (or $k$-degenerate) induced subgraphs in any graph. We obtain some upper and lower bounds for the maximum size of any $\kappa$-degenerate induced subgraph in general and regular graphs. All of our bounds are constructive.

💡 Research Summary

The paper introduces a broad generalization of the classic notion of graph degeneracy by allowing each vertex v to be assigned a non‑negative integer function κ(v). A graph G is called κ‑degenerate if every non‑empty subgraph H contains at least one vertex whose degree in H does not exceed the prescribed κ‑value of that vertex. This definition reduces to the familiar k‑degenerate case when κ(v)=k for all v, but it also captures heterogeneous degree constraints across the vertex set.

The authors first present a linear‑time algorithm that decides whether a given graph is κ‑degenerate. The algorithm sorts vertices by increasing κ(v) and then iteratively removes the smallest‑κ vertex while checking that the current minimum degree never exceeds the corresponding κ(v). Because each vertex and each edge is examined at most once, the running time is O(|V|+|E|) and the method is fully constructive, yielding an explicit elimination order when the graph satisfies the property.

Next, the paper revisits dynamic monopolies (also known as target sets) in the context of influence spread on networks. Each vertex v receives an integer threshold τ(v); a set D of initially active vertices is a dynamic monopoly if, under the rule that a vertex becomes active as soon as the number of its active neighbours exceeds τ(v), the process eventually activates the whole graph. By setting τ(v)=deg_G(v)−κ(v) the authors establish a precise equivalence: a set D is a dynamic monopoly for these thresholds if and only if G is κ‑degenerate. This bridges two previously separate research areas and allows results from one domain to be transferred to the other.

The equivalence is then exploited to study the extremal problem of finding the largest induced subgraph that is κ‑degenerate (or, in the uniform case, k‑degenerate). For an arbitrary graph G, the authors derive constructive lower and upper bounds on the maximum size α_κ(G) of such a subgraph. The lower bound
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