A Note on the Edge Roman Domination in Trees

A subset $X$ of edges of a graph $G$ is called an \textit{edgedominating set} of $G$ if every edge not in $X$ is adjacent tosome edge in $X$. The edge domination number $\gamma'(G)$ of $G$ is the minimum cardinality taken over all edge dominating sets of $G$. An \textit{edge Roman dominating function} of a graph $G$ is a function $f : E(G)\rightarrow \{0,1,2 \}$ such that every edge$e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e') = 2.$The weight of an edge Roman dominating function $f$ is the value$w(f)=\sum_{e\in E(G)}f(e)$. The edge Roman domination number of $G$, denoted by $\gamma_R'(G)$, is the minimum weight of an edge Roman dominating function of $G$. In this paper, we characterize trees with edge Roman domination number twice the edge domination number

Note on the upper bound of the rainbow index of a graph

A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is a rainbow path if every two edges of it receive distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the minimum number of colors that are needed to color the edges of $G$ such that there exists a rainbow path connecting every two vertices of $G$. Similarly, a tree in $G$ is a rainbow~tree if no two edges of it receive the same color. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow tree connecting $S$ for each $k$-subset $S$ of $V(G)$ is called the $k$-rainbow index of $G$, denoted by $rx_k(G)$, where $k$ is an integer such that $2\leq k\leq n$. Chakraborty et al. got the following result: For every $\epsilon> 0$, a connected graph with minimum degree at least $\epsilon n$ has bounded rainbow connection, where the bound depends only on $\epsilon$. Krivelevich and Yuster proved that if $G$ has $n$ vertices and the minimum degree $\delta(G)$ then $rc(G)<20n/\delta(G)$. This bound was later improved to $3n/(\delta(G)+1)+3$ by Chandran et al. Since $rc(G)=rx_2(G)$, a natural problem arises: for a general $k$ determining the true behavior of $rx_k(G)$ as a function of the minimum degree $\delta(G)$. In this paper, we give upper bounds of $rx_k(G)$ in terms of the minimum degree $\delta(G)$ in different ways, namely, via Szemer\'{e}di's Regularity Lemma, connected $2$-step dominating sets, connected $(k-1)$-dominating sets and $k$-dominating sets of $G$.Comment: 12 pages. arXiv admin note: text overlap with arXiv:0902.1255 by other author

Distributed algorithms for edge dominating sets

An edge dominating set for a graph G is a set D of edges such that each edge of G is in D or adjacent to at least one edge in D. This work studies deterministic distributed approximation algorithms for finding minimum-size edge dominating sets. The focus is on anonymous port-numbered networks: there are no unique identifiers, but a node of degree d can refer to its neighbours by integers 1, 2, ..., d. The present work shows that in the port-numbering model, edge dominating sets can be approximated as follows: in d-regular graphs, to within 4 − 6/(d + 1) for an odd d and to within 4 − 2/d for an even d; and in graphs with maximum degree Δ, to within 4 − 2/(Δ − 1) for an odd Δ and to within 4 − 2/Δ for an even Δ. These approximation ratios are tight for all values of d and Δ: there are matching lower bounds.Peer reviewe

On Strong Split Middle Domination of a Graph

The middle graph M(G) of graph G is obtained by inserting a vertex xi in the “middle” of each edge ei, 1 ? i ? |E(G)|, and adding the edge xixj for 1?i ? j ? |E(G)| if and only if ei and ej have a common vertex. A dominating set D of graph G is said to be a strong split dominating set of G if ?V(G) – D? is totally disconnected with at least two vertices. Strong split domination number is the minimum cardinality taken over all strong split dominating sets of G. In this paper we initiate the study of strong split middle domination of a graph. The strong split middle domination number of a graph G, denoted as ?ssm (G) is the minimum cardinality of strong split dominating set of M(G). In this paper many bounds on ?ssm(G) are obtained in terms of other domination parameters and elements of graph G. Also some equalities for ?ssm(G) are established

On Edge Dominating Number of Tensor Product of Cycle and Path

A subset S' of E(G) is called an edge dominating set ofG if every edge not in S' is adjacent to some edge in S'. The edge dominatingnumber of G, denoted by γ'(G), of G is the minimum cardinality takenover all edge dominating sets of G. Let G1 (V1, E1) and G2(V2,E2) betwo connected graph. The tensor product of G1 and G2, denoted byG1⨂▒G2 is a graph with the cardinality of vertex |V| = |V1| × |V2|and two vertices (u1,u2) and (v1,v2) in V are adjacent in G1⨂▒G2ifu1 v1 ∈ E1 and u2,v2 ∈E2 . In this paper we study an edge dominatingnumber in the tensor product of path and cycle. The results show thatγ'(Cn⨂▒P2) = ⌈2n/3⌉ for n is odd, γ'(Cn⨂▒P3) = n for n is odd, and theedge dominating number is undefined if n is even. For n ∈even number,we investigated the edge dominating number of its component on tensorproduct of cycle Cn and path. The results are γ'c(Cn⨂▒P2)= ⌈n/3⌉ andγ'c(Cn ⨂▒P3) = ⌈n/2⌉ which Cn ,P2 and P3, respectively, is Cycle order n,Path order 2 and Path order 3