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We consider extension variants of the classical graph problems Vertex Cover and Independent Set. Given a graph $G=(V,E)$ and a vertex set $U subseteq V$, it is asked if there exists a minimal vertex cover (resp. maximal independent set) $S$ with $Usubseteq S$ (resp. $U supseteq S$). Possibly contradicting intuition, these problems tend to be NP-hard, even in graph classes where the classical problem can be solved in polynomial time. Yet, we exhibit some graph classes where the extension variant remains polynomial-time solvable. We also study the parameterized complexity of these problems, with parameter $|U|$, as well as the optimality of simple exact algorithms under the Exponential-Time Hypothesis. All these complexity considerations are also carried out in very restricted scenarios, be it degree or topological restrictions (bipartite, planar or chordal graphs). This also motivates presenting some explicit branching algorithms for degree-bounded instances. We further discuss the price of extension, measuring the distance of $U$ to the closest set that can be extended, which results in natural optimization problems related to extension problems for which we discuss polynomial-time approximability.
Let $G$ be a graph on $n$ vertices and $mathrm{STAB}_k(G)$ be the convex hull of characteristic vectors of its independent sets of size at most $k$. We study extension complexity of $mathrm{STAB}_k(G)$ with respect to a fixed parameter $k$ (analogous
In this paper, we consider the Target Set Selection problem: given a graph and a threshold value $thr(v)$ for any vertex $v$ of the graph, find a minimum size vertex-subset to activate s.t. all the vertices of the graph are activated at the end of th
We study the generalized min sum set cover (GMSSC) problem, wherein given a collection of hyperedges $E$ with arbitrary covering requirements $k_e$, the goal is to find an ordering of the vertices to minimize the total cover time of the hyperedges; a
There are several centrality measures that have been introduced and studied for real world networks. They account for the different vertex characteristics that permit them to be ranked in order of importance in the network. Betweenness centrality is
For a given class $mathcal{C}$ of graphs and given integers $m leq n$, let $f_mathcal{C}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in any graph belonging to $mathcal{C}$ have a (possibly partial) rainbow independent $m$