A Lower Bound on the Double Outer-independent Domination Number of a Tree
نویسندگان
چکیده
A vertex of a graph is said to dominate itself and all of its neighbors. A double outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D, and the set V (G) \D is independent. The double outer-independent domination number of a graph G, denoted by γ d (G), is the minimum cardinality of a double outer-independent dominating set of G. We prove that for every nontrivial tree T of order n, with l leaves and s support vertices we have γ d (T ) ≥ (2n+ l−s+2)/3, and we characterize the trees attaining this lower bound. We also give a constructive characterization of trees T such that γ d (T ) = (2n+ 2)/3.
منابع مشابه
Bounds on the outer-independent double Italian domination number
An outer-independent double Italian dominating function (OIDIDF)on a graph $G$ with vertex set $V(G)$ is a function$f:V(G)longrightarrow {0,1,2,3}$ such that if $f(v)in{0,1}$ for a vertex $vin V(G)$ then $sum_{uin N[v]}f(u)geq3$,and the set $ {uin V(G)|f(u)=0}$ is independent. The weight ofan OIDIDF $f$ is the value $w(f)=sum_{vin V(G)}f(v)$. Theminimum weight of an OIDIDF on a graph $G$ is cal...
متن کاملOuter independent Roman domination number of trees
A Roman dominating function (RDF) on a graph G=(V,E) is a function f : V → {0, 1, 2} such that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. An RDF f is calledan outer independent Roman dominating function (OIRDF) if the set ofvertices assigned a 0 under f is an independent set. The weight of anOIRDF is the sum of its function values over ...
متن کاملOn trees with equal Roman domination and outer-independent Roman domination numbers
A Roman dominating function (RDF) on a graph $G$ is a function $f : V (G) to {0, 1, 2}$satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least onevertex $v$ for which $f(v) = 2$. A Roman dominating function $f$ is called an outer-independentRoman dominating function (OIRDF) on $G$ if the set ${vin Vmid f(v)=0}$ is independent.The (outer-independent) Roman dom...
متن کاملOn trees attaining an upper bound on the total domination number
A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$. The total domination number of a graph $G$, denoted by $gamma_t(G)$, is~the minimum cardinality of a total dominating set of $G$. Chellali and Haynes [Total and paired-domination numbers of a tree, AKCE International ournal of Graphs and Combinatorics 1 (2004), 6...
متن کاملOn the outer independent 2-rainbow domination number of Cartesian products of paths and cycles
Let G be a graph. A 2-rainbow dominating function (or 2-RDF) of G is a function f from V(G) to the set of all subsets of the set {1,2} such that for a vertex v ∈ V (G) with f(v) = ∅, thecondition $bigcup_{uin N_{G}(v)}f(u)={1,2}$ is fulfilled, wher NG(v) is the open neighborhoodof v. The weight of 2-RDF f of G is the value$omega (f):=sum _{vin V(G)}|f(v)|$. The 2-rainbowd...
متن کامل