2019
4
1
0
77
Strong Alliances in Graphs
2
2
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 $VS$ 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 1strong alliance number in terms of different graphical parameters are determined and the characterizations of graphs with 1strong alliance number 1, 2, and $n$ are obtained.
1

1
13


C.
Hegde
Mangalore University
Mangalore University
India
chandrugh@gmail.com


B.
Sooryanarayana
Dr. Ambedkar Institute of Technology
Dr. Ambedkar Institute of Technology
India
dr_bsnrao@drait.org
Alliances
Defensive alliances
Secure sets
Strong alliances
New skew equienergetic oriented graphs
2
2
Let $S(G^{sigma})$ be the skewadjacency 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 iftheir skew energies are equal. In this paper, we determine the skew spectra of some new oriented graphs. As applications, we give somenew methods to construct new noncospectral skew equienergetic oriented graphs.
1

15
24


Xiangxiang
Liu
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi'an, Shaanxi 710072,
People's Republic
of China
Department of Applied Mathematics, School
China
xxliumath@163.com


Ligong
Wang
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi&#039;an, Shaanxi 710072, People&#039;s Republic of China.
Department of Applied Mathematics, School
China
lgwangmath@163.com


Cunxiang
Duan
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi'an, Shaanxi 710072,
People's Republic
of China
Department of Applied Mathematics, School
China
cxduanmath@163.com
Oriented graph
Skew energy
Skew equienergetic
Eternal msecurity subdivision numbers in graphs
2
2
An eternal $m$secure set of a graph $G = (V,E)$ is aset $S_0subseteq V$ that can defend against any sequence ofsinglevertex attacks by means of multipleguard shifts along theedges of $G$. A suitable placement of the guards is called aneternal $m$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 wedelete the edge $uv$ from $G$ and add a new vertex $x$ and twoedges $ux$ and $vx$. The eternal $m$security subdivision number${rm sd}_{sigma_m}(G)$ of a graph $G$ is the minimum cardinalityof 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 showthat the eternal $m$security subdivision number of trees is atmost 2 and we characterize all trees attaining this bound.
1

25
33


Maryam
Atapour
Department of Mathematics
Faculty of basic sciences
University of Bonab
Bonab, Iran, Po. Box: 5551761167
Department of Mathematics
Faculty of basic
Iran
m.atapour@bonabu.ac.ir
eternal $m$secure set
eternal security number
eternal msecurity subdivision number
On the inverse maximum perfect matching problem under the bottlenecktype Hamming distance
2
2
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 bottlenecktype 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.
1

35
46


Javad
Tayyebi
Birjand university of technology
Birjand university of technology
Iran
javadtayyebi@birjand.ac.ir
Inverse problem
Hamming distance
perfect matching
binary search
The Roman domination and domatic numbers of a digraph
2
2
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 inneighbor $u$ 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.
1

47
59


Zhihong
Xie
College of Science, East China University of Technology, Nanchang, P. R. China
College of Science, East China University
China
xiezh168@163.com


Guoliang
Hao
College of Science, East China University of Technology, Nanchang, P. R. China
College of Science, East China University
China
guolianghao@163.com


Shouliu
Wei
Department of Mathematics, Minjiang University, Fuzhou, China
Department of Mathematics, Minjiang University,
China
wslwillow@126.com
Roman dominating function
Roman domination number
Roman domatic number
digraph
The Italian domatic number of a digraph
2
2
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 inneighborsassigned 1 under $f$ or one inneighbor $w$ with $f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinctItalian 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 anItalian 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 sharpbounds for $d_{I}(D)$. In addition, we determine the Italian domatic number of some digraphs.
1

61
70


Lutz
Volkmann
RWTH Aachen University
RWTH Aachen University
Germany
volkm@math2.rwthaachen.de
Digraphs
Italian dominating function
Italian domination number
Italian domatic number
On independent domination numbers of grid and toroidal grid directed graphs
2
2
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.
1

71
77


Ramy
Shaheen
ٍSyrian
ٍSyrian
Syria
shaheenramy2010@hotmail.com
directed path
directed cycle
Cartesian product
independent domination number