arXiv subjects

M. Chellali

Publications and source records attributed to M. Chellali.

7 recordsLinked to original sources

arXiv2026

Locating-dominating coalitions in graphs

A set $D$ of vertices in a graph $G = (V, E)$ is a locating-dominating set (LD-set) if it is dominating and every two vertices $u$, $v$ of $V\setminus D$ satisfy $N(u) \cap D \neq N(v) \cap D$. Two disjoint sets $A,B\subset V(G)$ form a locating-dominating coalition (for short, an LD-coalition) in $G$ if none of them is an LD-set in $G$ but their union $A\cup B$ is an LD-set. A locating-dominating coalition partition (for short, an LDC-partition) is a vertex partition $\Pi$ such that every set of $\Pi$ is not an LD-set in $G,$ but forms an LD-coalition with another set of $\Pi$. The locating-domination coalition number of $G$, denoted by $C_{L}(G),$ equals the maximum cardinality of an LDC-partition of $G$. Our purpose in this paper is to initiate the study of locating-dominating coalitions in graphs. We first investigate the existence of LDC-partitions. We also obtain lower and upper bounds on $C_{L}(G)$. We characterize connected graphs $G$ of order $n\ge 3$ satisfying $C_L(G) = n,$ as well as those trees $T$ such that $C_L(T)=n-1$. In addition, we determine the exact values of $C_L(G)$ for some classes of graphs. Moreover, we investigate the computational complexity of the decision problem associated with locating-dominating coalition partitions. To the best of our knowledge, this is the first work that addresses the algorithmic complexity of a decision problem related to coalition partitions, not only for this locating-dominating model but for coalition partitions in general.

math.CO↗

arXiv2024

Total restrained coalitions in graphs

A set $S\subseteq V$ in an isolate-free graph $G$ is a total restrained dominating set, abbreviated TRD-set, if every vertex in $V$ is adjacent to a vertex in $S$, and every vertex in $V\setminus S$ is adjacent to a vertex in $V\setminus S$. A total restrained coalition is made up of two disjoint sets of vertices $X$ and $Y$ of $G$, neither of which is a TRD-set but their union $X\cup Y$ is a TRD-set. A total restrained coalition partition of a graph $G$ is a partition $\Phi=\{V_1, V_2,\dots,V_k\}$ such that for all $i \in [k]$, the set $V_i$ forms a total restrained coalition with another set $V_j$ for some $j$, where $j\in [k]\setminus{i}$. The total restrained coalition number $C_{tr}(G)$ in $G$ equals the maximum order of a total restrained coalition partition in $G$. In this work, we initiate the study of total restrained coalition in graphs and its properties.

math.CO↗

arXiv2024

Total Roman {2}-Dominating functions in Graphs

A Roman $\{2\}$-dominating function (R2F) is a function $f:V\rightarrow \{0,1,2\}$ with the property that for every vertex $v\in V$ with $f(v)=0$ there is a neighbor $u$ of $v$ with $f(u)=2$, or there are two neighbors $x,y$ of $v$ with $f(x)=f(y)=1$. A total Roman $\{2\}$-dominating function (TR2DF) is an R2F $f$ such that the set of vertices with $f(v)>0$ induce a subgraph with no isolated vertices. The weight of a TR2DF is the sum of its function values over all vertices, and the minimum weight of a TR2DF of $G$ is the total Roman $\{2\}$-domination number $\gamma_{tR2}(G).$ In this paper, we initiate the study of total Roman $\{2\}$-dominating functions, where properties are established. Moreover, we present various bounds on the total Roman $\{2\}$-domination number. We also show that the decision problem associated with $\gamma_{tR2}(G)$ is NP-complete for bipartite and chordal graphs. {Moreover, we show that it is possible to compute this parameter in linear time for bounded clique-width graphs (including tres).}

math.CO↗

arXiv2024

On the outer independent total double Roman dominating functions

Let $\{0,1,\dots, t\}$ be abbreviated by $[t].$ A double Roman dominating function (DRDF) on a graph $\Gamma=(V,E)$ is a map $l:V\rightarrow [3]$ satisfying \textrm{(i)} if $l(r)=0$ then there must be at least two neighbors labeled 2 under $l$ or a neighbor $r'$ with $l(r')=3$; and \textrm{(ii)} if $l(r)=1$ then $r$ must be adjacent to a vertex $r'$ such that $l(r')\geq2$. A DRDF is an outer-independent total double Roman dominating function (OITDRDF) on $\Gamma$ if the set of vertices labeled $0$ induces an edgeless subgraph and the subgraph induced by the vertices with a non-zero label has no isolated vertices. The weight of an OITDRDF is the sum of its map values over all vertices, and the outer independent total Roman dominating number $\gamma_{tdR}^{oi}(\Gamma)$ is the minimum weight of an OITDRDF on $\Gamma$. First, we prove that the problem of determining $\gamma _{tdR}^{oi}(\Gamma)$ is NP-complete for bipartite and chordal graphs, after that, we prove that it is solvable in linear time when we are restricting to bounded clique-width graphs. Moreover, we present some tight bounds on $\gamma _{tdR}^{oi}(\Gamma)$ as well as the exact values for several graph families.

math.CO↗

arXiv2024

Maximal double Roman domination in graphs

A maximal double Roman dominating function (MDRDF) on a graph $G=(V,E)$ is a function $f:V(G)\rightarrow \{0,1,2,3\}$ such that \textrm{(i) }every vertex $v$ with $f(v)=0$ is adjacent to least two vertices { assigned $2$ or to at least one vertex assigned $3,$} \textrm{(ii) }every vertex $v$ with $f(v)=1$ is adjacent to at least one { vertex assigned $2$ or $3$} and \textrm{(iii) }the set $\{w\in V|~f(w)=0\}$ is not a dominating set of $G $. The weight of a MDRDF is the sum of its function values over all vertices, and the maximal double Roman domination number $\gamma _{dR}^{m}(G) $ is the minimum weight of an MDRDF on $G$. {In this paper, we initiate the study of maximal double Roman domination. We first show that the problem of determining }$\gamma _{dR}^{m}(G)$ {is NP-complete for bipartite, chordal and planar graphs. But it is solvable in linear time for bounded clique-width graphs including trees, cographs and distance-hereditary graphs. Moreover, we establish various relationships relating }$\gamma _{dR}^{m}(G)$ to some domination parameters. {For the class of trees, we show that for every tree }$T$ {of order }$n\geq 4,$ $\gamma _{dR}^{m}(T)\leq \frac{5}{4}n$ {and we characterize all trees attaining the bound. Finally, the exact values of }$\gamma _{dR}^{m}(G) $ {are given for paths and cycles.

math.CO↗

arXiv2021

Roman domination in graphs with minimum degree at least two and some forbidden cycles

Let $G=(V,E)$ be a graph of order $n$ and let $\gamma _{R}(G)$ and $\partial (G)$ denote the Roman domination number and the differential of $G,$ respectively. In this paper we prove that for any integer $k\geq 0$, if $G$ is a graph of order $n\geq 6k+9$, minimum degree $\delta \geq 2,$ which does not contain any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$% -cycles, then $\gamma _{R}(G)\leq \frac{(4k+8)n}{6k+11}$. This bound is an improvement of the bounds given in [E.W. Chambers, B. Kinnersley, N. Prince, and D.B. West, Extremal problems for Roman domination, SIAM J. Discrete Math. 23 (2009) 1575--1586] when $k=0,$ {and [S. Bermudo, On the differential and Roman domination number of a graph with minimum degree two, Discrete Appl. Math. 232 (2017), 64--72] when }$k=1.$ Moreover, using the Gallai-type result involving the Roman domination number and the differential of graphs established by Bermudo et al. stating that $\gamma _{R}(G)+\partial (G)=n$, we have $\partial (G)\geq \frac{(2k+3)n}{6k+11},$ thereby settling the conjecture of Bermudo posed in the second paper.

math.CO↗

arXiv2021

Restrained condition on double Roman dominating functions

We continue the study of restrained double Roman domination in graphs. For a graph $G=\big{(}V(G),E(G)\big{)}$, a double Roman dominating function $f$ is called a restrained double Roman dominating function (RDRD function) if the subgraph induced by $\{v\in V(G)\mid f(v)=0\}$ has no isolated vertices. The restrained double Roman domination number (RDRD number) $\gamma_{rdR}(G)$ is the minimum weight $\sum_{v\in V(G)}f(v)$ taken over all RDRD functions of $G$. We first prove that the problem of computing $\gamma_{rdR}$ is NP-hard even for planar graphs, but it is solvable in linear time when restricted to bounded clique-width graphs such as trees, cographs and distance-hereditary graphs. Relationships between $\gamma_{rdR}$ and some well-known parameters such as restrained domination number $\gamma_{r}$, domination number $\gamma$ and restrained Roman domination number $\gamma_{rR}$ are investigated in this paper by bounding $\gamma_{rdR}$ from below and above involving $\gamma_{r}$, $\gamma$ and $\gamma_{rR}$ for general graphs, respectively. We prove that $\gamma_{rdR}(T)\geq n+2$ for any tree $T\neq K_{1,n-1}$ of order $n\geq2$ and characterize the family of all trees attaining the lower bound. The characterization of graphs with small RDRD numbers is given in this paper.

math.CO↗