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Given a graph $G=(V,E)$ and an integer $k ge 1$, a $k$-hop dominating set $D$ of $G$ is a subset of $V$, such that, for every vertex $v in V$, there exists a node $u in D$ whose hop-distance from $v$ is at most $k$. A $k$-hop dominating set of minimum cardinality is called a minimum $k$-hop dominating set. In this paper, we present linear-time algorithms that find a minimum $k$-hop dominating set in unicyclic and cactus graphs. To achieve this, we show that the $k$-dominating set problem on unicycle graph reduces to the piercing circular arcs problem, and show a linear-time algorithm for piercing sorted circular arcs, which improves the best known $O(nlog n)$-time algorithm.
Let $P$ be a path graph of $n$ vertices embedded in a metric space. We consider the problem of adding a new edge to $P$ so that the radius of the resulting graph is minimized, where any center is constrained to be one of the vertices of $P$. Previous
For any $epsilon>0$, Laue and Matijevi{c} [CCCG07, IPL08] give a PTAS for finding a $(1+epsilon)$-approximate solution to the $k$-hop MST problem in the Euclidean plane that runs in time $(n/epsilon)^{O(k/epsilon)}$. In this paper, we present an algo
In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph $D$ on $n$ vertices and $m$ edges, and an integer $k$. The objective is to determine whether there exists a set of at most $k$ vertices intersecting every directed cycl
Given an undirected, weighted graph, the minimum spanning tree (MST) is a tree that connects all of the vertices of the graph with minimum sum of edge weights. In real world applications, network designers often seek to quickly find a replacement edg
Dealing with the NP-complete Dominating Set problem on undirected graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar gra