Domination Number Of Graph Research Articles

Bounds on the signed distance-݇ -domination number of graphs

Let 􀜩 􀵌 􁈺􀜸, 􀜧􁈻 be a graph with vertex set 􀜸 􀵌 􀜸􁈺􀜩􁈻 of order 􀝊 and edge set 􀜧 􀵌 􀜧􁈺􀜩􁈻. A 􀝇 -dominating set of 􀜩 is a subset 􀜵 􀙃 􀜸 such that each vertex in 􀜸 􀜵 has at least 􀝇 neighbors in 􀜵. If 􀝒 is a vertex of a graph 􀜩, the open 􀝇-neighborhood of 􀝒, denoted by 􀜰􀯞􁈺􀝒􁈻, is the set 􀜰􀯞􁈺􀝒􁈻 􀵌 􁈼􀝑 􀗐 􀜸 􀗷 􀝑 􀵍 􀝒 􀜽􀝊􀝀 􀝀􁈺􀝑, 􀝒􁈻 􀵑 􀝇 􁈽. 􀜰􀯞􁈾􀝒􁈿 􀵌 􀜰􀯞􁈺􀝒􁈻 􀗫 􁈼􀝒􁈽 is the closed 􀝇-neighborhood of 􀝒. A function 􀝂 􀗷 􀜸 􀸷 􁈼􀵆1, 1􁈽 is a signed distance-􀝇 dominating function of 􀜩, if for every vertex 􀝒 􀗐 􀜸, 􀝂􁈺􀜰􀯞􁈾􀝒􁈿􁈻 􀵌 Σ􀯨 􀗐 􀯇􀳖􁈾􀯩􁈿 􀝂􁈺􀝑􁈻 􀵒 1. The signed distance-􀝇-domination number, denoted by 􀟛􀯞,􀯦􁈺􀜩􁈻, is the minimum weight of a signed distance-􀝇-dominating function of 􀜩. In this paper, we give lower and upper bounds on 􀟛􀯞,􀯦 of graphs. Also, we determine the signed distance-􀝇-domination number of graph 􀟛􀯞,􀯦􁈺􀜩 􀗩 􀜪􁈻 (the graph obtained from the disjoint union 􀜩 􀵅 􀜪 by adding the edges 􁈼􀝔􀝕 􀗷 􀝔 􀗐 􀜸􁈺􀜩􁈻, 􀝕 􀗐 􀜸􁈺􀜪􁈻􁈽) when 􀝇 􀵒 2.

On the ratio of the domination number and the independent domination number in graphs

We let γ(G) and i(G) denote the domination number and the independent domination number of G, respectively. Recently, Rad and Volkmann conjectured that i(G)/γ(G)≤Δ(G)/2 for every graph G, where Δ(G) is the maximum degree of G. In this note, we construct counterexamples of the conjecture for Δ(G)≥6 and give a sharp upper bound of the ratio i(G)/γ(G) by using the maximum degree of G.

Relations between the Roman k-domination and Roman domination numbers in graphs

Let G = (V, E) be a graph and let k be a positive integer. A Roman k-dominating function ( R k-DF) on G is a function f : V(G) → {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices v1, v2, …, vk with f(vi) = 2 for i = 1, 2, …, k. The weight of an R k-DF is the value f(V(G)) = ∑u∈V(G) f(u) and the minimum weight of an R k-DF on G is called the Roman k-domination number γkR(G) of G. In this paper, we present relations between γkR(G) and γR(G). Moreover, we give characterizations of some classes of graphs attaining equality in these relations. Finally, we establish a relation between γkR(G) and γR(G) for {K1,3, K1,3+e}-free graphs and we characterize all such graphs G with γkR(G) = γR(G)+t, where [Formula: see text].

Formula omitted]-domination in graphs

A vertex subset D of a graph G=(V,E) is a [1,2]-set if, 1≤|N(v)∩D|≤2 for every vertex v∈V∖D, that is, each vertex v∈V∖D is adjacent to either one or two vertices in D. The minimum cardinality of a [1,2]-set of G, denoted by γ[1,2](G), is called the [1,2]-domination number of G. Chellali et al. (2013) showed that there exist graphs G of order n with γ[1,2](G)=n. But, the complete characterization of such graphs seems to be a difficult task. As responding to some open questions posed by Chellali et al., we further show that such graphs exist even among some special families of graphs, such as planar graphs, bipartite graphs (triangle-free graphs). It is also shown that for a tree T of order n≥3 with k leaves, if dT(v)≥4 for any non-leaf vertex v, then γ[1,2](T)=n−k. In addition, Nordhaus–Gaddum-type inequalities are established for the [1,2]-domination numbers of graphs. Thereby, we solve several open problems posed by Chellali et al.

Global minus domination in graphs

‎A function $f:V(G)rightarrow {-1,0,1}$ is a minus‎ ‎dominating function if for every vertex $vin V(G)$‎, ‎$sum_{uin‎ ‎N[v]}f(u)ge 1$‎. ‎A minus dominating function $f$ of $G$ is called‎ ‎a global minus dominating function if $f$ is also a minus‎ ‎dominating function of the complement $overline{G}$ of $G$‎. ‎The‎ global minus domination number $gamma_{g}^-(G)$ of $G$ is‎ ‎defined as $gamma_{g}^-(G)=min{sum_{vin V(G)} f(v)mid f‎ ‎;{rm is; a; global; minus; dominating; function} {rm of‎ ‎}; G}$‎. ‎In this paper we initiate the study of the global minus ‎domination number in graphs and we establish lower and upper‎ ‎bounds for the global minus domination number‎.

Signed Roman edge domination numbers in graphs

The closed neighborhood $$N_G[e]$$NG[e] of an edge $$e$$e in a graph $$G$$G is the set consisting of $$e$$e and of all edges having a common end-vertex with $$e$$e. Let $$f$$f be a function on $$E(G)$$E(G), the edge set of $$G$$G, into the set $$\{-1, 1, 2\}$${-1,1,2}. If $$ \sum _{x\in N[e]}f(x) \ge 1$$?x?N[e]f(x)?1 for every edge $$e$$e of $$G$$G and every edge $$e$$e for which $$f (e) = -1$$f(e)=-1 is adjacent to at least one edge $$e'$$e? for which $$f (e')= 2$$f(e?)=2, then $$f$$f is called a signed Roman edge dominating function of $$G$$G. The minimum of the values $$\sum _{e\in E(G)} f(e)$$?e?E(G)f(e), taken over all signed Roman edge dominating functions $$f$$f of $$G$$G, is called the signed Roman edge domination number of $$G$$G and is denoted by $$\gamma _{sR}'(G)$$?sR?(G). In this note we initiate the study of the signed Roman edge domination in graphs and present some (sharp) bounds for this parameter.

A new lower bound for the total domination number in graphs proving a Graffiti.pc Conjecture

The total domination number γt(G) of a graph G is the minimum cardinality of a set S of vertices so that every vertex of G is adjacent to a vertex in S. Our main result of this paper is that if G is a connected graph and x1,x2,x3∈V(G), then γt(G)≥14(d(x1,x2)+d(x1,x3)+d(x2,x3)). Furthermore if equality holds in this bound, then the multiset {d(x1,x2),d(x1,x3),d(x2,x3)} is equal to {2,3,3} modulo four. As a consequence of this result, we prove a conjecture on the total domination number. To state this conjecture, let B denote the set of vertices of maximum eccentricity in G and let ecc(B) denote the maximum distance in G of a vertex outside B to a vertex of B. The following conjecture is known as Graffiti.pc Conjecture #233 (http://cms.dt.uh.edu/faculty/delavinae/research/wowII): if G is a connected graph of order at least two, then γt(G)≥2(ecc(B)+1)/3. We prove this conjecture. In fact, as a consequence of our main result stated earlier we prove the following much stronger result: if G is a connected graph of order at least two, then γt(G)≥(3ecc(B)+2)/4. We also prove that if G is a connected graph and x1,x2,x3,x4∈V(G), then γt(G)≥18(d(x1,x2)+d(x1,x3)+d(x1,x4)+d(x2,x3)+d(x2,x4)+d(x3,x4)), and this result is best possible.

Bounds on the Size of the Minimum Dominating Sets of Some Cylindrical Grid Graphs

Let γPm □ Cn denote the domination number of the cylindrical grid graph formed by the Cartesian product of the graphs Pm, the path of length m, m≥2, and the graph Cn, the cycle of length n, n≥3. In this paper we propose methods to find the domination numbers of graphs of the form Pm □ Cn with n≥3 and m=5 and propose tight bounds on domination numbers of the graphs P6 □ Cn, n≥3. Moreover, we provide rough bounds on domination numbers of the graphs Pm □ Cn, n≥3 and m≥7. We also point out how domination numbers and minimum dominating sets are useful for wireless sensor networks.

LIMITED DOMINATING BROADCAST IN GRAPHS

For a given graph G, a function f : V(G) → {0, 1, …, diamG} such that for every vertex v of G, f(v) ≤ e(v) (where e(v) is the eccentricity of the vertex v and diamG is the diameter of G), is a broadcast on G. A broadcast f is a dominating broadcast if Nf[V+] = V(G), where V+ = {u | f(u) > 0}, Nf[V+] = ⋃u∈V+Nf[u], and Nf[u] = {v | d(u, v) ≤ f(u)}. In this paper, inspired by Dunbar et al. [3], we study limited dominating broadcast in graphs and obtain some properties, bounds, and characterizations for limited broadcast domination number in graphs.

-Tuple Total Restrained Domination in Complementary Prisms

In a graph with , a -tuple total restrained dominating set is a subset of such that each vertex of is adjacent to at least vertices of and also each vertex of is adjacent to at least vertices of . The minimum number of vertices of such sets in is the -tuple total restrained domination number of . In [-tuple total restrained domination/domatic in graphs, BIMS], the author initiated the study of the -tuple total restrained domination number in graphs. In this paper, we continue it in the complementary prism of a graph.

Some New Perspectives on Global Domination in Graphs

A dominating set is called a global dominating set if it is a dominating set of a graph G and its complement G¯. Here we explore the possibility to relate the domination number of graph G and the global domination number of the larger graph obtained from G by means of various graph operations. In this paper we consider the following problem: Does the global domination number remain invariant under any graph operations? We present an affirmative answer to this problem and establish several results.

A note on the independent domination number in graphs

We obtain an upper bound for the independent domination number of a graph in terms of the domination number and maximum degree.

Global vertex-edge domination chain and its characterization

In agraph G, a subset S of V is Global vertex-edge dominating set if S is vertex-edge dominating set in both G and . In this paper we have introduced new concepts such as Global vertex-edge dominating set, Global vertex-edge irredundant set, Global independent vertex-edge domination set. Also we have identified the Global vertex-edge domination chain, An attempt is made to characterize the Global vertex-edge domination number of graphs. We have identified the graphs whose Global vertex-edge domination number is 1, 2, 3 & k.

A note on domination and independence-domination numbers of graphs

Vizing's conjecture is true for graphs G satisfying γ i ( G ) = γ ( G ), where γ ( G ) is the domination number of a graph G and γ i ( G ) is the independence-domination number of G , that is, the maximum, over all independent sets I in G , of the minimum number of vertices needed to dominate I . The equality γ i ( G ) = γ ( G ) is known to hold for all chordal graphs and for chordless cycles of length 0 (mod 3). We prove some results related to graphs for which the above equality holds. More specifically, we show that the problems of determining whether γ i ( G ) = γ ( G ) = 2 and of verifying whether γ i ( G ) ≥ 2 are NP-complete, even if G is weakly chordal. We also initiate the study of the equality γ i = γ in the context of hereditary graph classes and exhibit two infinite families of graphs for which γ i < γ .

Three conjectures on the signed cycle domination in graphs

Let G=(V,E) be a graph, a function g:E→{−1,1} is said to be a signed cycle dominating function (SCDF for short) of G if ∑e∈E(C)g(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as γsc(G)=min{∑e∈E(G)g(e)∣g is an SCDF of G}. Xu (Discrete Math. 309:1007–1012, 2009) first researched the signed cycle domination number of graphs and raised the following conjectures: (1) Let G be a maximal planar graphs of order n≥3. Then γsc(G)=n−2; (2) For any graph G with δ(G)=3, γsc(G)≥1; (3) For any 2-connected graph G, γsc(G)≥1. In this paper, we present some results about these conjectures.

Domination Number of Graphs Without Small Cycles

It has been shown (J. Harant and D. Rautenbach, Domination in bipartite graphs. Discrete Math. 309:113–122, 2009) that the domination number of a graph of order n and minimum degree at least 2 that does not contain cycles of length 4, 5, 7, 10 nor 13 is at most $${\frac{3n}{8}}$$. They believed that the assumption that the graphs do not contain cycles of length 10 might be replaced by the exclusion of finitely many exceptional graphs. In this paper, we positively answer that if G is a connected graph of order n and minimum degree at least 2 that does not contain cycles of length 4, 5 nor 7 and is not one of three exceptional graphs, then the domination number of G is at most $${\frac{3n}{8}}$$.

A New Bound on the Domination Number of Graphs with Minimum Degree Two

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

Outer-k-connected component domination in graphs

A subset S of the vertices of a graph G is an outer-connected dominating set, if S is a dominating set of G and G — S is connected. The outer-connected domination number of G, denoted by , is the minimum cardinality of an OCDS of G. In this paper we generalize the outer-connected domination in graphs. Many of the known results and bounds of outer-connected domination number are immediate consequences of our results. Let k ≥ 1 be an integer. A subset D of vertices of G is an outer-k-connected component dominating set if D is a dominating and the graph G — D has exactly k connected component. The outer-k-connected component domination number of G, denoted by , is the minimum cardinality of a outer-k-connected component dominating set of G. We study the outer-k-connected component domination in a graph G. We present properties and bounds of outer-k-connected component domination number in graphs, and show that the decision problem for the outer-k-connected component domination number of an arbitrary graph G is NP-complete. Finally, we determine for several certain classes of graphs G.

Bounds on the connected [formula omitted]-domination number in graphs

Let G = ( V , E ) be a simple graph, and let k be a positive integer. A subset D ⊆ V is a k - dominating set of the graph G if every vertex v ∈ V − D is adjacent to at least k vertices of D . The k - domination number γ k ( G ) is the minimum cardinality among the k -dominating sets of G . A subset D ⊆ V is said to be a connected k - dominating set if D is k -dominating and its induced subgraph is connected. D is called total k - dominating if every vertex in V has at least k neighbors in D and it is a connected total k - dominating set if, additionally, its induced subgraph is connected. The minimum cardinalities of a connected k -dominating set, a total k -dominating set, and a connected total k -dominating set are respectively denoted as γ k c ( G ) , γ k t ( G ) and γ k c , t ( G ) . In this paper, we establish different sharp bounds on the connected k -domination number γ k c ( G ) , involving also the parameters γ k ( G ) , γ k t ( G ) and γ k c , t ( G ) .

BOUNDS ON THE DOMINATION NUMBER OF PERMUTATION GRAPHS

For a graph G with n vertices and a permutation α on V(G), a permutation graph Pα(G) is obtained from two identical copies of G by adding an edge between v and α(V) for any v ϵ V(G). Let γ(G) be the domination number of a graph G. It has been shown that γ(G) ≤ γ(Pα(G) ≤ 2γ(G) for any permutation α on V(G). In this paper, we investigate specific graphs for which there exists a permutation α such that γ(Pα(G)) ≻ γ(G) in terms of the domination number of G or the maximum degree of G. Additionally, we construct a class of graphs for which the domination number of any permutation graph is twice the domination number of the original graph, as well as explore finding a specific graph G and permutation α for any two positive integers a and b with a ≤ b ≤ 2a, to have γ(G) = a and γ(Pα(G)) = b.