A dominating set $S$ is an Isolate Dominating Set (IDS) if the induced subgraph $G[S]$ has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate dominating set $S\subseteq V$ is an isolate secure dominating set (ISDS), if for each vertex $u \in V \setminus S$, there exists a neighboring vertex $v$ of $u$ in $S$ such that $(S \setminus \{v\}) \cup \{u\}$ is an IDS of $G$. The minimum cardinality of an ISDS of $G$ is called as an isolate secure domination number, and is denoted by $γ_{0s}(G)$. Given a graph $ G=(V,E)$ and a positive integer $ k,$ the ISDM problem is to check whether $ G $ has an isolate secure dominating set of size at most $ k.$ We prove that ISDM is NP-complete even when restricted to bipartite graphs and split graphs. We also show that ISDM can be solved in linear time for graphs of bounded tree-width.
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…
Let $G = (V,E)$ be a simple, undirected and connected graph. A connected dominating set $S \subseteq V$ is a secure connected dominating set of $G$, if for each $ u \in V\setminus S$, there exists $v\in S$ such that $(u,v) \in E$ and the set $(S \setminus \{ v \}) \cup \{ u \} $ is a connected dominating set of $G$. The minimum size of a secure connected dominating set of $G$ denoted by $ γ_{sc} (G)$, is called the secure connected domination number of $G$. Given a graph $ G$ and a positive integer $ k,$ the Secure Connected Domination (SCDM) problem is to check whether $ G $ has a secure conn…
A set $S \subseteq V$ is a dominating set in G if for every u \in V \ S, there exists $v \in S$ such that $(u,v) \in E$, i.e., $N[S] = V$. A dominating set $S$ is an Isolate Dominating Set} (IDS) if the induced subgraph $G[S]$ has at least one isolated vertex. It is known that Isolate Domination Decision problem (IDOM) is NP-complete for bipartite graphs. In this paper, we extend this by showing that the IDOM is NP-complete for split graphs and perfect elimination bipartite graphs, a subclass of bipartite graphs. A set $S \subseteq V$ is an independent set if G[S] has no edge. A set S \subsete…
Let $G(V,E)$ be a simple, undirected and connected graph. A dominating set $S \subseteq V(G)$ is called a $2$-\textit{secure dominating set} ($2$-SDS) in $G$, if for every pair of distinct vertices $u_1,u_2 \in V(G)$ there exists a pair of distinct vertices $v_1,v_2 \in S$ such that $v_1 \in N[u_1]$, $v_2 \in N[u_2]$ and $(S \setminus \{v_1,v_2\}) \cup \{u_1,u_2 \}$ is a dominating set in $G$. The $2$\textit{-secure domination number} denoted by $γ_{2s}(G)$, equals the minimum cardinality of a $2$-SDS in $G$. Given a graph $ G$ and a positive integer $ k,$ the $ 2 $-Secure Domination ($ 2 $-SD…
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}…
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 $|V(G)|\leq 16$.
In a graph $G=(V,E)$ with no isolated vertex, a dominating set $D \subseteq V$, is called a semitotal dominating set if for every vertex $u \in D$ there is another vertex $v \in D$, such that distance between $u$ and $v$ is at most two in $G$. Given a graph $G=(V,E)$ without isolated vertices, the Minimum Semitotal Domination problem is to find a minimum cardinality semitotal dominating set of $G$. The semitotal domination number, denoted by $γ_{t2}(G)$, is the minimum cardinality of a semitotal dominating set of $G$. The decision version of the problem remains NP-complete even when restricted…
For a simple graph $G=(V,E)$ without any isolated vertex, a cosecure dominating set $D$ of $G$ satisfies the following two properties (i) $S$ is a dominating set of $G$, (ii) for every vertex $v \in S$ there exists a vertex $u \in V \setminus S$ such that $uv \in E$ and $(S \setminus \{v\}) \cup \{u\}$ is a dominating set of $G$. The minimum cardinality of a cosecure dominating set of $G$ is called cosecure domination number of $G$ and is denoted by $γ_{cs}(G)$. The Minimum Cosecure Domination problem is to find a cosecure dominating set of a graph $G$ of cardinality $γ_{cs}(G)$. The decision …
For a graph $G=(V,E)$ with no isolated vertices, a set $D\subseteq V$ is called a semipaired dominating set of G if $(i)$ $D$ is a dominating set of $G$, and $(ii)$ $D$ can be partitioned into two element subsets such that the vertices in each two element set are at distance at most two. The minimum cardinality of a semipaired dominating set of $G$ is called the semipaired domination number of $G$, and is denoted by $γ_{pr2}(G)$. The \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $γ_{pr2}(G)$. In this paper, we initiate the algorithm…
A set $S$ of vertices in a graph $G$ is a dominating set if every vertex not in $S$ is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The independent domination number $i(G)$ of $G$ is the minimum cardinality of an independent dominating set in $G$. In 2013 Goddard and Henning [Discrete Math 313 (2013), 839--854] conjectured that if $G$ is a connected cubic graph of order $n$, then $i(G) \le \frac{3}{8}n$, except if $G$ is the complete bipartite graph $K_{3,3}$ or the $5$-prism $C_5 \, \Box \, K_2$. Further they construct two…
A set $D\subseteq V$ of a graph $G=(V,E)$ is called a neighborhood total dominating set of $G$ if $D$ is a dominating set and the subgraph of $G$ induced by the open neighborhood of $D$ has no isolated vertex. Given a graph $G$, \textsc{Min-NTDS} is the problem of finding a neighborhood total dominating set of $G$ of minimum cardinality. The decision version of \textsc{Min-NTDS} is known to be \textsf{NP}-complete for bipartite graphs and chordal graphs. In this paper, we extend this \textsf{NP}-completeness result to undirected path graphs, chordal bipartite graphs, and planar graphs. We also…
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…
