scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

scholarly journals A linear time algorithm for minimum equitable dominating set in trees

2021 ◽  
Vol 40 (4) ◽  
pp. 805-814
Author(s):  
Sohel Rana ◽  
Sk. Md. Abu Nayeem
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Minimum Cardinality ◽  
Np Complete ◽  
General Graphs

Let G = (V, E) be a graph. A subset De of V is said to be an equitable dominating set if for every v ∈ V \ De there exists u ∈ De such that uv ∈ E and |deg(u) − deg(v)| ≤ 1, where, deg(u) and deg(v) denote the degree of the vertices u and v respectively. An equitable dominating set with minimum cardinality is called the minimum equitable dominating set and its cardinality is called the equitable domination number and it is denoted by γe. The problem of finding minimum equitable dominating set in general graphs is NP-complete. In this paper, we give a linear time algorithm to determine minimum equitable dominating set of a tree.

scholarly journals Semipaired Domination in Some Subclasses of Chordal Graphs

2021 ◽  
Vol vol. 23 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Michael A. Henning ◽  
Arti Pandey ◽  
Vikash Tripathi
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Minimum Cardinality ◽  
Positive Side ◽  
Domination Problem ◽  
Np Complete ◽  
Decision Version

A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.

scholarly journals On the L-Grundy domination number of a graph

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3205-3215 ◽  
Author(s):  
Bostjan Bresar ◽  
Tanja Gologranc ◽  
Michael Henning ◽  
Tim Kos
Keyword(s):  
Lower Bound ◽  
Regular Graph ◽  
Decision Problem ◽  
Linear Time ◽  
Domination Number ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Zero Forcing ◽  
Split Graph ◽  
Np Complete

In this paper, we continue the study of the L-Grundy domination number of a graph introduced and first studied in [Grundy dominating sequences and zero forcing sets, Discrete Optim. 26 (2017) 66-77]. A vertex in a graph dominates itself and all vertices adjacent to it, while a vertex totally dominates another vertex if they are adjacent. A sequence of distinct vertices in a graph G is called an L-sequence if every vertex v in the sequence is such that v dominates at least one vertex that is not totally dominated by any vertex that precedes v in the sequence. The maximum length of such a sequence is called the L-Grundy domination number, L gr(G), of G. We show that the L-Grundy domination number of every forest G on n vertices equals n, and we provide a linear-time algorithm to find an L-sequence of length n in G. We prove that the decision problem to determine if the L-Grundy domination number of a split graph G is at least k for a given integer k is NP-complete. We establish a lower bound on L gr(G) when G is a regular graph, and investigate graphs G on n vertices for which L gr(G) = n.

scholarly journals On k-rainbow domination in middle graphs

2021 ◽  
Author(s):  
Kijung Kim
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Matching Number ◽  
Domatic Number ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Middle Graph ◽  
Dominating Function

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$. The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$. The \textit{$k$-rainbow domination number} $\gamma_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$. In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs. We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it and determine the middle $3$-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number. In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.

scholarly journals Some progress on the mixed Roman domination in graphs

2020 ◽  
Author(s):  
Hossein Abdollahzadeh Ahangar ◽  
Jafar Amjadi ◽  
Mustapha Chellali ◽  
S. Kosari ◽  
Vladimir Samodivkin ◽  
...  
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function ◽  
Np Complete ◽  
Domination In Graphs ◽  
Dominating Function

Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. A mixed Roman dominating function (MRDF) of $G$ is a function $f:V\cup E\rightarrow \{0,1,2\}$ satisfying the condition that every element $x\in V\cup E$ for which $f(x)=0$ is adjacent or incident to at least one element $% y\in V\cup E$ for which $f(y)=2$. The weight of a mixed Roman dominating function $f$ is $\omega (f)=\sum_{x\in V\cup E}f(x)$. The mixed Roman domination number $\gamma _{R}^{\ast }(G)$ of $G$ is the minimum weight of a mixed Roman dominating function of $G$. We first show that the problem of computing $\gamma _{R}^{\ast }(G)$ is NP-complete for bipartite graphs and then we present upper and lower bounds on the mixed Roman domination number, some of them are for the class of trees.

Isolate Roman domination in graphs

2021 ◽  
pp. 2150131
Author(s):  
Davood Bakhshesh
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Chordal Graphs ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function ◽  
Np Complete ◽  
Domination In Graphs ◽  
Dominating Function

Let [Formula: see text] be a graph with the vertex set [Formula: see text]. A function [Formula: see text] is called a Roman dominating function of [Formula: see text], if every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text]. The weight of a Roman dominating function [Formula: see text] is equal to [Formula: see text]. The minimum weight of a Roman dominating function of [Formula: see text] is called the Roman domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we initiate the study of a variant of Roman dominating functions. A function [Formula: see text] is called an isolate Roman dominating function of [Formula: see text], if [Formula: see text] is a Roman dominating function and there is a vertex [Formula: see text] with [Formula: see text] which is not adjacent to any vertex [Formula: see text] with [Formula: see text]. The minimum weight of an isolate Roman dominating function of [Formula: see text] is called the isolate Roman domination number of [Formula: see text], denoted by [Formula: see text]. We present some upper bound on the isolate Roman domination number of a graph [Formula: see text] in terms of its Roman domination number and its domination number. Moreover, we present some classes of graphs [Formula: see text] with [Formula: see text]. Finally, we show that the decision problem associated with the isolate Roman dominating functions is NP-complete for bipartite graphs and chordal graphs.

Sign in / Sign up

Export Citation Format

Share Document