A dominating set of a graph $G$ is a set $S \subseteq V(G)$ such that every vertex in $V(G) \setminus S$ has a neighbor in $S$, where two vertices are neighbors if they are adjacent. A secure dominating set of $G$ is a dominating set $S$ of $G$ with the additional property that for every vertex $v \in V(G) \setminus S$, there exists a neighbor $u$ of $v$ in $S$ such that $(S \setminus \{u\}) \cup \{v\}$ is a dominating set of $G$. The secure domination number of $G$, denoted by $γ_s(G)$, is the minimum cardinality of a secure dominating set of $G$. We prove that if $G$ is a $P_5$-free graph, then $γ_s(G) \le \frac{3}{2}α(G)$, where $α(G)$ denotes the independence number of $G$. We further show that if $G$ is a connected $(P_5, H)$-free graph for some $H \in \{ P_3 \cup P_1, K_2 \cup 2K_1, ~\text{paw},~ C_4\}$, then $γ_s(G)\le \max\{3,α(G)\}$. We also show that if $G$ is a $(P_3 \cup P_2)$-free graph, then $γ_s(G)\le α(G)+1$.
Let $G = (V,E)$ be a simple, undirected and connected graph. A connected (total) dominating set $S \subseteq V$ is a secure connected (total) dominating set of $G$, if for each $ u \in V \setminus S$, there exists $v \in S$ such that $uv \in E$ and $(S \setminus \lbrace v \rbrace) \cup \lbrace u \rbrace $ is a connected (total) dominating set of $G$. The minimum cardinality of a secure connected (total) dominating set of $G$ denoted by $ γ_{sc} (G) (γ_{st}(G))$, is called the secure connected (total) domination number of $G$. In this paper, we show that the decision problems corresponding to s…
The secure domination numbers of the Cartesian products of two small graphs with paths or cycles is determined, as well as for Mobius ladder graphs. Prior to this work, in all cases where the secure domination number has been determined, the proof has either been trivial, or has been derived from lower bounds established by considering different forms of domination. However, the latter mode of proof is not applicable for most graphs, including those considered here. Hence, this work represents the first attempt to determine secure domination numbers via the properties of secure domination itse…
The secure domination problem, a variation of the domination problem with some important real-world applications, is considered. Very few algorithmic attempts to solve this problem have been presented in literature, and the most successful to date is a binary programming formulation which is solved using CPLEX. A new binary programming formulation is proposed here which requires fewer constraints and fewer binary variables than the existing formulation. It is implemented in CPLEX, and tested on certain families of graphs that have previously been considered in the context of secure domination.…
Let $w=(w_0,w_1, \dots,w_l)$ be a vector of nonnegative integers such that $ w_0\ge 1$. Let $G$ be a graph and $N(v)$ the open neighbourhood of $v\in V(G)$. We say that a function $f: V(G)\longrightarrow \{0,1,\dots ,l\}$ is a $w$-dominating function if $f(N(v))=\sum_{u\in N(v)}f(u)\ge w_i$ for every vertex $v$ with $f(v)=i$. The weight of $f$ is defined to be $ω(f)=\sum_{v\in V(G)} f(v)$. Given a $w$-dominating function $f$ and any pair of adjacent vertices $v, u\in V(G)$ with $f(v)=0$ and $f(u)>0$, the function $f_{u\rightarrow v}$ is defined by $f_{u\rightarrow v}(v)=1$, $f_{u\rightarrow v}…
Jha, Pradhan, and Banerjee devised a linear algorithm to compute the secure domination number of a cograph. Here it is shown that their Lemma~2, which is crucial for the computational complexity of the algorithm, is incomplete. An accordingly modified lemma is proved and it is demonstrated that the complexity of the modified algorithm remains linear.
Given a graph G = (V,E), a subset S of V is dominating if for every v in V - S there exists u in S such that uv is in E. A dominating subset S of V is secure if for every v in V - S there exists u in S such that (S - {u}) U {v} is dominating. In this work we extend the concept of secure dominating set to digraphs in four different ways, all of them with interesting applications, and prove some results regarding each of them.
In this paper we study the weak Roman domination number and the secure domination number of a graph. In particular, we obtain general bounds on these two parameters and, as a consequence of the study, we derive new inequalities of Nordhaus-Gaddum type involving secure domination and weak Roman domination. Furthermore, the particular case of Cartesian product graphs is considered.
In this note, we consider the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by using only one edge contraction? We show that the problem is polynomial-time solvable on $P_3+kP_2$-free graphs for any $k \geq 0$ which combined with results of [1,2] leads to a complexity dichotomy of the problem on $H$-free graphs.
In 1975, Plesník characterized all triangle-free planar graphs as having a diameter $2$. We characterize all triangle-free projective-planar graphs having a diameter $2$ and discuss some applications. In particular, the main result is applied to calculate the analogue of clique numbers for graphs, namely, colored mixed graphs, having different types of arcs and edges.
As a fundamental research object, the minimum edge dominating set (MEDS) problem is of both theoretical and practical interest. However, determining the size of a MEDS and the number of all MEDSs in a general graph is NP-hard, and it thus makes sense to find special graphs for which the MEDS problem can be exactly solved. In this paper, we study analytically the MEDS problem in the pseudofractal scale-free web and the Sierpiński gasket with the same number of vertices and edges. For both graphs, we obtain exact expressions for the edge domination number, as well as recursive solutions to the n…
In a graph $G$, an {\it efficient dominating set} is a subset $D$ of vertices such that $D$ is an independent set and each vertex outside $D$ has exactly one neighbor in $D$. The {\textsc{Efficient Dominating Set}} problem (EDS) asks for the existence of an efficient dominating set in a given graph $G$. The EDS is known to be $NP$-complete for $P_7$-free graphs, and is known to be polynomial time solvable for $P_5$-free graphs. However, the computational complexity of the EDS problem is unknown for $P_6$-free graphs. In this paper, we show that the EDS problem can be solved in polynomial time …
We study the relationship between the eternal domination number of a graph and its clique covering number using both large-scale computation and analytic methods. In doing so, we answer two open questions of Klostermeyer and Mynhardt. We show that the smallest graph having its eternal domination number less than its clique covering number has $10$ vertices. We determine the complete set of $10$-vertex and $11$-vertex graphs having eternal domination numbers less than their clique covering numbers. We show that the smallest triangle-free graph with this property has order $13$, as does the smal…
In a graph $G$, an efficient dominating set is a subset $D$ of vertices such that $D$ is an independent set and each vertex outside $D$ has exactly one neighbor in $D$. The Minimum Weight Efficient Dominating Set (Min-WED) problem asks for an efficient dominating set of total minimum weight in a given vertex-weighted graph; the Maximum Weight Efficient Dominating Set (Max-WED) problem is defined similarly. The Min-WED/Max-WED is known to be $NP$-complete for $P_7$-free graphs, and is known to be polynomial time solvable for $P_5$-free graphs. However, the computational complexity of the Min-WE…
The $k$-power domination problem is a problem in graph theory, which has applications in many areas. However, it is hard to calculate the exact $k$-power domination number since determining k-power domination number of a generic graph is a NP-complete problem. We determine the exact $k$-power domination number in two graphs which have the same number of vertices and edges: pseudofractal scale-free web and Sierpiński gasket. The $k$-power domination number becomes 1 for $k\ge2$ in the Sierpiński gasket, while the $k$-power domination number increases at an exponential rate with regard to the nu…
A dominating set $D$ in a digraph is a set of vertices such that every vertex is either in $D$ or has an in-neighbour in $D$. A dominating set $D$ of a digraph is locating-dominating if every vertex not in $D$ has a unique set of in-neighbours within $D$. The location-domination number $γ_L(G)$ of a digraph $G$ is the smallest size of a locating-dominating set of $G$. We investigate upper bounds on $γ_L(G)$ in terms of the order of $G$. We characterize those digraphs with location-domination number equal to the order or the order minus one. Such digraphs always have many twins: vertices with t…
In this paper, we first show that the power domination number of a connected $4$-regular claw-free graph on $n$ vertices is at most $\frac{n+1}{5}$, and the bound is sharp. The statement partly disprove the conjecture presented by Dorbec et al. in SIAM J. Discrete Math., 27:1559-1574, 2013. Then we present a dynamic programming style linear-time algorithm for weighted power domination problem in trees.
In this paper, we give a series of couterexamples to negate a conjecture and hence answer an open question on the $k$-power domination of regular graphs (see [P. Dorbec et al., SIAM J. Discrete Math., 27 (2013), pp. 1559-1574]). Furthermore, we focus on the study of $k$-power domination of claw-free graphs. We show that for $l\in\{2,3\}$ and $k\ge l$, the $k$-power domination number of a connected claw-free $(k+l+1)$-regular graph on $n$ vertices is at most $\frac{n}{k+l+2}$, and this bound is tight.
We show that every connected induced subgraph of a graph $G$ is dominated by an induced connected split graph if and only if $G$ is $\cal{C}$-free, where $\cal{C}$ is a set of six graphs which includes $P_7$ and $C_7$, and each containing an induced $P_5$. A similar characterisation is shown for the class of graphs which are dominated by induced complete split graphs. Motivated by these results, we study structural descriptions of some classes of $\cal{C}$-free graphs. In particular, we give structural descriptions for the class of ($P_7$,$C_7$,$C_4$,gem)-free graphs and for the class of ($P_7…
Addressing a problem posed by Chellali, Haynes, and Hedetniemi (Discrete Appl. Math. 178 (2014) 27-32) we prove $γ_{r2}(G)\leq 2γ_r(G)$ for every graph $G$, where $γ_{r2}(G)$ and $γ_r(G)$ denote the $2$-rainbow domination number and the weak Roman domination number of $G$, respectively. We characterize the extremal graphs for this inequality that are $\{ K_4,K_4-e\}$-free, and show that the recognition of the $K_5$-free extremal graphs is NP-hard.
In any graph $G$, the domination number $γ(G)$ is at most the independence number $α(G)$. The Inverse Domination Conjecture says that, in any isolate-free $G$, there exists pair of vertex-disjoint dominating sets $D, D'$ with $|D|=γ(G)$ and $|D'| \leq α(G)$. Here we prove that this statement is true if the upper bound $α(G)$ is replaced by $\frac{3}{2}α(G) - 1$ (and $G$ is not a clique). We also prove that the conjecture holds whenever $γ(G)\leq 5$ or $|V(G)|\leq 16$.
For a graph $G=(V,E)$, a subset $D$ of vertex set $V$, is a dominating set of $G$ if every vertex not in $D$ is adjacent to atleast one vertex of $D$. A dominating set $D$ of a graph $G$ with no isolated vertices is called a paired dominating set (PD-set), if $G[D]$, the subgraph induced by $D$ in $G$ has a perfect matching. The Min-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the Min-PD problem remains NP-complete even when $G$ belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the positive side, the problem is effic…
Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique covering numbers of graph are explored in this paper. Among other results, we characterize bipartite and triangle-free graphs with domination and eternal domination numbers equal to two, trees with equal …
Let $γ_g(G)$ be the game domination number of a graph $G$. Rall conjectured that if $G$ is a traceable graph, then $γ_g(G) \le \left\lceil \frac{1}{2}n(G)\right\rceil$. Our main result verifies the conjecture over the class of line graphs. Moreover, in this paper we put forward the conjecture that if $δ(G) \geq 2$, then $γ_g(G) \leq \left\lceil \frac{1}{2}n(G) \right\rceil$. We show that both conjectures hold true for claw-free cubic graphs. We further prove the upper bound $γ_g(G) \le \left\lceil \frac{11}{20} \, n(G) \right\rceil$ over the class of claw-free graphs of minimum degree at least…
Let $γ_g(G)$ and $γ_{tg}(G)$ be the game domination number and the total game domination number of a graph $G$, respectively. Then $G$ is $γ_g$-perfect (resp. $γ_{tg}$-perfect), if every induced subgraph $F$ of $G$ satisfies $γ_g(F)=γ(F)$ (resp. $γ_{tg}(F)=γ_t(F)$). A recursive characterization of $γ_g$-perfect graphs is derived. The characterization yields a polynomial recognition algorithm for $γ_g$-perfect graphs. It is proved that every minimally $γ_g$-imperfect graph has domination number $2$. All minimally $γ_g$-imperfect triangle-free graphs are determined. It is also proved that $γ_{tg…
A graph $G$ is said to be $k$-$γ$-vertex critical if the domination numbers $γ(G)$ of $G$ is $k$ and $γ(G - v) < k$ for any vertex $v$ of $G$. Similarly, A graph $G$ is said to be $k$-$γ_{c}$-vertex critical if the connected domination numbers $γ_{c}(G)$ of $G$ is $k$ and $γ_{c}(G - v) < k$ for any vertex $v$ of $G$. The problem of interest is to determine whether or not $2$-connected $k$-$γ$-vertex critical graphs are Hamiltonian. In this paper, for all $k \geq 3$, we provide a $2$-connected $k$-$γ$-vertex critical graph which is non-Hamiltonian. We prove that every $2$-connected $3$-$γ$-vert…
We propose the conjecture that the domination number $γ(G)$ of a $Δ$-regular graph $G$ with $Δ\geq 1$ is always at most its edge domination number $γ_e(G)$, which coincides with the domination number of its line graph. We prove that $γ(G)\leq \left(1+\frac{2(Δ-1)}{Δ2^Δ}\right)γ_e(G)$ for general $Δ\geq 1$, and $γ(G)\leq \left(\frac{7}{6}-\frac{1}{204}\right)γ_e(G)$ for $Δ=3$. Furthermore, we verify our conjecture for cubic claw-free graphs.
The power domination problem focuses on finding the optimal placement of phase measurement units (PMUs) to monitor an electrical power network. In the context of graphs, the power domination number of a graph $G$, denoted $γ_P(G)$, is the minimum number of vertices needed to observe every vertex in the graph according to a specific set of observation rules. In \cite{ZKC_cubic}, Zhao et al. proved that if $G$ is a connected claw-free cubic graph of order $n$, then $γ_P(G) \leq n/4$. In this paper, we show that if $G$ is a claw-free diamond-free cubic graph of order $n$, then $γ_P(G) \le n/6$, a…
Let $G$ be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x) =\sum d(G, i)x^i, where d(G,i) is the number of dominating sets of G of size i. Every root of D(G,x) is called the domination root of G. It is clear that (0,\infty) is zero free interval for domination polynomial of a graph. It is interesting to investigate graphs which have complex domination roots with positive real parts. In this paper, we first investigate complexity of the domination polynomial at specific points. Then we present and investigate some families of graphs whose complex domination ro…
It is conjectured that the game domination number is at most $3n/5$ for every $n$-vertex graph which does not contain isolated vertices. It was proved in the recent years that the conjecture holds for several graph classes, including the class of forests and that of graphs with minimum degree at least two. Here we prove that the slightly bigger upper bound $5n/8$ is valid for every isolate-free graph.
Given a simple, finite graph with vertex set $V(G)$, we define a zero forcing set of $G$ as follows. Choose $S\subseteq V(G)$ and color all vertices of $S$ blue and all vertices in $V(G) - S$ white. The color change rule is if $w$ is the only white neighbor of blue vertex $v$, then we change the color of $w$ from white to blue. If after applying the color change rule as many times as possible eventually every vertex of $G$ is blue, we call $S$ a zero forcing set of $G$. $Z(G)$ denotes the minimum cardinality of a zero forcing set. Davila and Henning proved in \cite{zerocubic} that for any claw…
Let $G$ be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $γ_t(G)$. A set $S$ of vertices in $G$ is a disjunctive total dominating set of $G$ if every vertex is adjacent to a vertex of $S$ or has at least two vertices in $S$ at distance2 from it. The disjunctive total domination number, $γ^d_t(G)$, is the minimum cardinality of such a set. We observe that $γ^d_t(G) \le γ_t(G)$. We prove that if $G$ is a connected graph of order$n \ge 8$, then $γ^d_t(G) \le 2(n-1)/…
In the domination game studied here, Dominator and Staller alternately choose a vertex of a graph $G$ and take it into a set $D$. The number of vertices dominated by the set $D$ must increase in each single turn and the game ends when $D$ becomes a dominating set of $G$. Dominator aims to minimize whilst Staller aims to maximize the number of turns (or equivalently, the size of the dominating set $D$ obtained at the end). Assuming that Dominator starts and both players play optimally, the number of turns is called the game domination number $γ_g(G)$ of $G$. Kinnersley, West and Zamani verified…
The aim of this work is to investigate the nonnegative signed domination number $γ^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $γ^{NN}_s$ for regular graphs and prove that $n/3$ is the best possible upper bound on this parameter for a cubic graph of order $n$, specifically. As an application of the classic theorem of Turán we bound $γ^{NN}_s(G)$ from below, for an ($r+1$)-clique-free graph $G$ and characterize all such graphs for which the equality holds, which corrects and generalizes a result for bipartite graphs in [Electron. J. …
The ratio of the connected domination number, $γ_c$, and the domination number, $γ$, is strictly bounded from above by 3. It was shown by Zverovich that for every connected $(P_5,C_5)$-free graph, $γ_c = γ$. In this paper, we investigate the interdependence of $γ$ and $γ_c$ in the class of $(P_k,C_k)$-free graphs, for $k \ge 6$. We prove that for every connected $(P_6,C_6)$-free graph, $γ_c \le γ+ 1$ holds, and there is a family of $(P_6,C_6)$-free graphs with arbitrarily large values of $γ$ attaining this bound. Moreover, for every connected $(P_8,C_8)$-free graph, $γ_c / γ\le 2$, and there i…
For a graph $G$, let $γ_R(G)$ and $γ_{r2}(G)$ denote the Roman domination number of $G$ and the $2$-rainbow domination number of $G$, respectively. It is known that $γ_{r2}(G)\leq γ_R(G)\leq \frac{3}{2}γ_{r2}(G)$. Fujita and Furuya (Difference between 2-rainbow domination and Roman domination in graphs, Discrete Applied Mathematics 161 (2013) 806-812) present some kind of characterization of the graphs $G$ for which $γ_R(G)-γ_{r2}(G)=k$ for some integer $k$. Unfortunately, their result does not lead to an algorithm that allows to recognize these graphs efficiently. We show that for every fixed…
Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The \emph{Dominating Induced Matching} (\emph{DIM}) problem asks for the existence of a d.i.m.\ in $G$; this problem is also known as the \emph{Efficient Edge Domination} problem; it is the Efficient Domination problem for line graphs. The DIM problem is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 but is solvable in linear time for $P_7$…
A dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex outside $D$ is adjacent to a vertex in $D$. A locating-dominating set of $G$ is a dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \neq N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-domination number of $G$, denoted $γ_L(G)$, is the …
A set of vertices $S$ in a simple isolate-free graph $G$ is a semi-total dominating set of $G$ if it is a dominating set of $G$ and every vertex of $S$ is within distance 2 or less with another vertex of $S$. The semi-total domination number of $G$, denoted by $γ_{t2}(G)$, is the minimum cardinality of a semi-total dominating set of $G$. In this paper, we study semi-total domination of Cartesian products of graphs. Our main result establishes that for any graphs $G$ and $H$, $γ_{t2}(G\,\square\, H)\ge \frac{1}{3}γ_{t2}(G)γ_{t2}(H)$.
The Z-domination game is a variant of the domination game in which each newly selected vertex $u$ in the game must have a not yet dominated neighbor, but after the move all vertices from the closed neighborhood of $u$ are declared to be dominated. The Z-domination game is the fastest among the five natural domination games. The corresponding game Z-domination number of a graph $G$ is denoted by $γ_{Zg}(G)$. It is proved that the game domination number and the game total domination number of a graph can be expressed as the game Z-domination number of appropriate lexicographic products. Graphs w…
For a graph $G$, a vertex subset $S \subseteq V(G)$ is said to be $K_{k}$-isolating if $G - N_{G}[S]$ does not contain $K_{k}$ as a subgraph. The $K_{k}$-isolation number of $G$, denoted by $ι_{k}(G)$, is the minimum cardinality of a $K_{k}$-isolating set of $G$. Analogously, $S$ is said to be independent $K_{k}$-isolating if $S$ is a $K_{k}$-isolating set of $G$ and $G[S]$ has no edge. The independent $K_{k}$-isolation number of $G$, denoted by $ι'_{k}(G)$, is the minimum cardinality of an independent $K_{k}$-isolating set of $G$. Clearly, when $k = 1$, we have $γ(G) = ι_{1}(G)$ and $i(G) = ι…
A set $D$ of vertices in $G$ is a disjunctive dominating set in $G$ if every vertex not in $D$ is adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it in $G$. The disjunctive domination number, $γ^{d}_2(G)$, of $G$ is the minimum cardinality of a disjunctive dominating set in $G$. In this paper, we show that if $G$ be a graph of order at least $3$, $δ(G)\geq 2$ and with no component isomorphic to any of eight forbidden graphs, then $γ^{d}_2(G)\leq \frac{|G|}{3}$. Moreover, we provide an infinite family of graphs attaining this bound. In addition, we also stud…
A vertex subset $S$ of a graph $G$ is a double dominating set of $G$ if $|N[v]\cap S|\geq 2$ for each vertex $v$ of $G$, where $N[v]$ is the set of the vertex $v$ and vertices adjacent to $v$. The double domination number of $G$, denoted by $γ_{\times 2}(G)$, is the cardinality of a smallest double dominating set of $G$. A graph $G$ is said to be double domination edge critical if $γ_{\times 2}(G+e)<γ_{\times 2}(G)$ for any edge $e \notin E$. A double domination edge critical graph $G$ with $γ_{\times 2}(G)=k$ is called $k$-$γ_{\times 2}(G)$-critical. A graph $G$ is $r$-factor-critical if $G-S…
