### فهرست مطالب

• Volume:4 Issue:1, 2019
• تاریخ انتشار: 1397/12/02
• تعداد عناوین: 7
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• CHANDRU HEGDE *, B Sooryanarayana Pages 1-13
For any simple connected graph $G=(V,E)$, a defensive alliance is a subset $S$ of $V$ satisfying the condition that every vertex $vin S$ has at most one more neighbour in $V-S$ than it has in $S$. The minimum cardinality of any defensive alliance in $G$ is called the alliance number of $G$, denoted $a(G)$. In this paper, we introduce a new type of alliance number called $k$-strong alliance number and its varieties. The bounds for 1-strong alliance number in terms of different graphical parameters are determined and the characterizations of graphs with 1-strong alliance number 1, 2, and $n$ are obtained.
Keywords: Alliances, Defensive alliances, Secure sets, Strong alliances
• Xiangxiang Liu, Ligong Wang *, Cunxiang Duan Pages 15-24
Let $S(G^{sigma})$ be the skew-adjacency matrix of the oriented graph $G^{sigma}$, which is obtained from a simple undirected graph $G$ by assigning an orientation $sigma$ to each of its edges. The skew energy of an oriented graph $G^{sigma}$ is defined as the sum of absolute values of all eigenvalues of $S(G^{sigma})$. Two oriented graphs are said to be skew equienergetic if their skew energies are equal. In this paper, we determine the skew spectra of some new oriented graphs. As applications, we give some new methods to construct new non-cospectral skew equienergetic oriented graphs.
Keywords: Oriented graph, Skew energy, Skew equienergetic
• Maryam Atapour * Pages 25-33
An eternal $m$-secure set of a graph $G = (V,E)$ is a set $S_0subseteq V$ that can defend against any sequence of single-vertex attacks by means of multiple-guard shifts along the edges of $G$. A suitable placement of the guards is called an eternal $m$-secure set. The eternal $m$-security number $sigma_m(G)$ is the minimum cardinality among all eternal $m$-secure sets in $G$. An edge $uvin E(G)$ is subdivided if we delete the edge $uv$ from $G$ and add a new vertex $x$ and two edges $ux$ and $vx$. The eternal $m$-security subdivision number ${rm sd}_{sigma_m}(G)$ of a graph $G$ is the minimum cardinality of a set of edges that must be subdivided (where each edge in $G$ can be subdivided at most once) in order to increase the eternal $m$-security number of $G$. In this paper, we study the eternal $m$-security subdivision number in trees. In particular, we show that the eternal $m$-security subdivision number of trees is at most 2 and we characterize all trees attaining this bound.
Keywords: eternal $m$-secure set, eternal -security number, eternal m-security subdivision number
• Javad Tayyebi * Pages 35-46
Given an undirected network G(V,A,c) and a perfect matching M of G, the inverse maximum perfect matching problem consists of modifying minimally the elements of c so that M becomes a maximum perfect matching with respect to the modified vector. In this article, we consider the inverse problem when the modifications are measured by the weighted bottleneck-type Hamming distance. We propose an algorithm based on the binary search technique for solving the problem. Our proposed algorithm has a better time complexity than the one presented in cite{Liu}. We also study the inverse assignment problem as a special case of the inverse maximum perfect matching problem in which the network is bipartite and present an efficient algorithm for solving the problem. Finally, we compare the algorithm with those presented in the literature.
Keywords: Inverse problem, Hamming distance, perfect matching, binary search
• Zhihong Xie, Guoliang Hao *, Shouliu Wei Pages 47-59
A Roman dominating function (RDF) on a digraph $D$ is a function $f: V(D)rightarrow {0,1,2}$ satisfying the condition that every vertex $v$ with $f(v)=0$ has an in-neighbor $u$ with $f(u)=2$. The weight of an RDF $f$ is the value $sum_{vin V(D)}f(v)$. The Roman domination number of a digraph $D$ is the minimum weight of an RDF on $D$. A set ${f_1,f_2,dots,f_d}$ of Roman dominating functions on $D$ with the property that $sum_{i=1}^df_i(v)le2$ for each $vin V(D)$, is called a Roman dominating family (of functions) on $D$. The maximum number of functions in a Roman dominating family on $D$ is the Roman domatic number of $D$, denoted by $d_{R}(D)$. In this paper we continue the investigation of the Roman domination number, and we initiate the study of the Roman domatic number in digraphs. We present some bounds for $d_{R}(D)$. In addition, we determine the Roman domatic number of some digraphs.
Keywords: Roman dominating function, Roman domination number, Roman domatic number, digraph
• Lutz Volkmann * Pages 61-70
An {em Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function $fcolon V(D)to {0, 1, 2}$ such that every vertex $vin V(D)$ with $f(v)=0$ has at least two in-neighbors assigned 1 under $f$ or one in-neighbor $w$ with $f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinct Italian dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vin V(D)$, is called an {em Italian dominating family} (of functions) on $D$. The maximum number of functions in an Italian dominating family on $D$ is the {em Italian domatic number} of $D$, denoted by $d_{I}(D)$. In this paper we initiate the study of the Italian domatic number in digraphs, and we present some sharp bounds for $d_{I}(D)$. In addition, we determine the Italian domatic number of some digraphs.
Keywords: Digraphs, Italian dominating function, Italian domination number, Italian domatic number
• Ramy Shaheen * Pages 71-77
A subset $S$ of vertex set $V(D)$ is an {\em indpendent dominating set} of $D$ if $S$ is both an independent and a dominating set of $D$. The {\em indpendent domination number}, $i(D)$ is the cardinality of the smallest independent dominating set of $D$.
In this paper we calculate the independent domination number of the { \em cartesian product} of two {\em directed paths} $P_m$ and $P_n$ for arbitraries $m$ and $n$. Also, we calculate the independent domination number of the { \em cartesian product} of two {\em directed cycles} $C_m$ and $C_n$ for $m, n \equiv 0 ({\rm mod}\ 3)$, and $n \equiv 0 ({\rm mod}\ m)$. There are many values of $m$ and $n$ such that $C_m \Box C_n$ does not have an independent dominating set.
Keywords: directed path, directed cycle, cartesian product, independent domination number