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Parameterized complexity of $r$-Hop, $r$-Step, and $r$-Hop Roman Domination

Sandip Das, Sweta Das, Sk Samim Islam

TL;DR

This paper examines the algorithmic complexity of several well-known exact-distance variants of domination, namely \textsc{$r$-Step Domination}, \textsc{$r$-Hop Domination}, and \textsc{$r$-Hop Roman Domination], and proves that for all $r\ge 2, the problems \textsc{$r$-Hop Roman Domination} is \textsc{W[2]}-complete.

Abstract

The \textsc{Dominating Set} problem is a classical and extensively studied topic in graph theory and theoretical computer science. In this paper, we examine the algorithmic complexity of several well-known exact-distance variants of domination, namely \textsc{$r$-Step Domination}, \textsc{$r$-Hop Domination}, and \textsc{$r$-Hop Roman Domination}. Let $G$ be a graph and let $r \geq 2$ be an integer. A set $S \subseteq V(G)$ is an \emph{$r$-hop dominating set} if every vertex in $V(G)\setminus S$ is at distance exactly $r$ from some vertex of $S$. Similarly, $S$ is an \emph{$r$-step dominating set} if every vertex of $G$ lies at distance exactly $r$ from at least one vertex of $S$. An \emph{$r$-hop Roman dominating function} on $G$ is a function $f \colon V(G)\to\{0,1,2\}$ such that for every vertex $v$ with $f(v)=0$, there exists a vertex $u$ at distance exactly $r$ from $v$ with $f(u)=2$. The \emph{weight} of $f$ is defined as $f(V)=\sum_{v\in V(G)} f(v)$. The \textsc{$r$-Hop Domination} (respectively, \textsc{$r$-Step Domination}) problem asks whether $G$ admits an $r$-hop dominating set (respectively, $r$-step dominating set) of size at most $k$, while the \textsc{$r$-Hop Roman Domination} problem asks whether $G$ admits an $r$-hop Roman dominating function of weight at most $k$. It is known that for every $r\ge 2$, the problems \textsc{$r$-Step Domination}, \textsc{$r$-Hop Domination}, and \textsc{$r$-Hop Roman Domination} are \textsc{NP}-complete. First we prove that for all $r\ge 2$, \textsc{$r$-Hop Roman Domination} is \textsc{W[2]}-complete. Furthermore, for every $r\ge 2$, \textsc{$r$-Step Domination} and \textsc{$r$-Hop Domination} remain \textsc{W[2]}-hard even when restricted to bipartite graphs and chordal graphs. Unless the ETH fails, none of these problems admits an algorithm running in time $2^{o(n+m)}$ on graphs with $n$ vertices and $m$ edges.

Parameterized complexity of $r$-Hop, $r$-Step, and $r$-Hop Roman Domination

TL;DR

This paper examines the algorithmic complexity of several well-known exact-distance variants of domination, namely \textsc{-Step Domination}, \textsc{-Hop Domination}, and \textsc{-Hop Roman Domination], and proves that for all r$-Hop Roman Domination} is \textsc{W[2]}-complete.

Abstract

The \textsc{Dominating Set} problem is a classical and extensively studied topic in graph theory and theoretical computer science. In this paper, we examine the algorithmic complexity of several well-known exact-distance variants of domination, namely \textsc{-Step Domination}, \textsc{-Hop Domination}, and \textsc{-Hop Roman Domination}. Let be a graph and let be an integer. A set is an \emph{-hop dominating set} if every vertex in is at distance exactly from some vertex of . Similarly, is an \emph{-step dominating set} if every vertex of lies at distance exactly from at least one vertex of . An \emph{-hop Roman dominating function} on is a function such that for every vertex with , there exists a vertex at distance exactly from with . The \emph{weight} of is defined as . The \textsc{-Hop Domination} (respectively, \textsc{-Step Domination}) problem asks whether admits an -hop dominating set (respectively, -step dominating set) of size at most , while the \textsc{-Hop Roman Domination} problem asks whether admits an -hop Roman dominating function of weight at most . It is known that for every , the problems \textsc{-Step Domination}, \textsc{-Hop Domination}, and \textsc{-Hop Roman Domination} are \textsc{NP}-complete. First we prove that for all , \textsc{-Hop Roman Domination} is \textsc{W[2]}-complete. Furthermore, for every , \textsc{-Step Domination} and \textsc{-Hop Domination} remain \textsc{W[2]}-hard even when restricted to bipartite graphs and chordal graphs. Unless the ETH fails, none of these problems admits an algorithm running in time on graphs with vertices and edges.
Paper Structure (7 sections, 20 theorems, 3 figures)

This paper contains 7 sections, 20 theorems, 3 figures.

Table of Contents

  1. Abstract
  2. Introduction
  3. Preliminaries
  4. $r$-Hop Roman Domination (Proof of Theorem \ref{['thm:combined_r_roman_hop']})
  5. $r$-Step Domination (Proof of Theorem \ref{['thm:$r$-step domination restricted']})
  6. $r$-Hop Domination (Proof of Theorem \ref{['thm:$r$-hop domination restricted']})
  7. Conclusion

Key Result

Theorem 1

Let $r \geq 2$. The $r$-Hop Roman Domination problem satisfies the following:

Figures (3)

  • Figure 1: This is a pictorial description of the gadget construction of $G'$ from $G$, for $4$-Hop Roman Domination problem.
  • Figure 2: This is a pictorial description of the gadget construction of $G'$ from $G$, for $5$-Hop Roman Domination problem.
  • Figure 3: This is a pictorial description of the gadget construction of $G_2$ from $G_1$.

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • Lemma 7
  • Lemma 8
  • Corollary 9
  • Corollary 10
  • Corollary 11
  • ...and 10 more