The concepts of critical and cocritical radius edge-invariant graphs are introduced. We prove that every graph can be embedded as an induced subgraph of a critical or cocritical radius-edge-invariant graph. We show that every cocritical radius-edge-invariant graph of radius r ≥ 15 must have at least 3r + 2 vertices.

A graph G is hereditarily dominated by a class D of connected graphs if each connected induced subgraph of G contains a dominating induced subgraph belonging to D. In this paper we characterize graphs hereditarily dominated by classes of complete bipartite graphs, stars, connected bipartite graphs, and complete k-partite graphs.

Subgraphs reveal information about the geometry and functionalities of complex networks. For scale-free networks with unbounded degree fluctuations, we obtain asymptotics number times a small connected graph occurs as subgraph or an induced subgraph. We these results by analyzing configuration model exponent τ∈(2,3) introducing novel class optimization problems. any given subgraph, unique optim...

LetG = (V,E) be a simple undirected graph. We call any subset C ⊆ V an identifying code if the sets I(v) = {c ∈ C | {v, c} ∈ E or v = c} are distinct and non-empty for all vertices v ∈ V . A graph is called twin-free if there is an identifying code in the graph. The identifying code with minimum size in a twin-free graph G is called the optimal identifying code and the size of such a code is de...

We show that the problems of deciding whether an arbitrary graph has a k-regular subgraph, for some given k 3; of deciding whether a planar graph has a quartic subgraph; and of deciding whether a planar graph has a quintic subgraph, are complete for NP via logspace reductions. There are no regular planar graphs of degree greater than 5.