Strong Alliances in Graphs
C.
Hegde
Mangalore University
author
B.
Sooryanarayana
Dr. Ambedkar Institute of Technology
author
text
article
2019
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
4
v.
1
no.
2019
1
13
http://comb-opt.azaruniv.ac.ir/article_13785_db97a57dfa7d3980c88f4ce8245a31b6.pdf
dx.doi.org/10.22049/cco.2018.25921.1056
New skew equienergetic oriented graphs
Xiangxiang
Liu
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi'an, Shaanxi 710072,
People's Republic
of China
author
Ligong
Wang
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi&#039;an, Shaanxi 710072, People&#039;s Republic of China.
author
Cunxiang
Duan
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi'an, Shaanxi 710072,
People's Republic
of China
author
text
article
2019
eng
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 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 non-cospectral skew equienergetic oriented graphs.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
4
v.
1
no.
2019
15
24
http://comb-opt.azaruniv.ac.ir/article_13786_f2e382fc36d6b753b4c6c9d7e3b92229.pdf
dx.doi.org/10.22049/cco.2018.26286.1093
Eternal m-security subdivision numbers in graphs
Maryam
Atapour
Department of Mathematics
Faculty of basic sciences
University of Bonab
Bonab, Iran, Po. Box: 5551761167
author
text
article
2019
eng
An eternal $m$-secure set of a graph $G = (V,E)$ is aset $S_0subseteq V$ that can defend against any sequence ofsingle-vertex attacks by means of multiple-guard 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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
4
v.
1
no.
2019
25
33
http://comb-opt.azaruniv.ac.ir/article_13803_3fd4e7d1ecc4a8bed4b5eb43305015eb.pdf
dx.doi.org/10.22049/cco.2018.25948.1058
On the inverse maximum perfect matching problem under the bottleneck-type Hamming distance
Javad
Tayyebi
Birjand university of technology
author
text
article
2019
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
4
v.
1
no.
2019
35
46
http://comb-opt.azaruniv.ac.ir/article_13804_dbb06f4741821880df74fe3c9cca70f8.pdf
dx.doi.org/10.22049/cco.2018.26231.1087
The Roman domination and domatic numbers of a digraph
Zhihong
Xie
College of Science, East China University of Technology, Nanchang, P. R. China
author
Guoliang
Hao
College of Science, East China University of Technology, Nanchang, P. R. China
author
Shouliu
Wei
Department of Mathematics, Minjiang University, Fuzhou, China
author
text
article
2019
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
4
v.
1
no.
2019
47
59
http://comb-opt.azaruniv.ac.ir/article_13841_9bb2c1e1cb7db1916a327f1866d037b2.pdf
dx.doi.org/10.22049/cco.2019.26356.1101
The Italian domatic number of a digraph
Lutz
Volkmann
RWTH Aachen University
author
text
article
2019
eng
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-neighborsassigned 1 under $f$ or one in-neighbor $w$ 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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
4
v.
1
no.
2019
61
70
http://comb-opt.azaruniv.ac.ir/article_13845_b207051edc37d9a82ecd605da8ed79b4.pdf
dx.doi.org/10.22049/cco.2019.26360.1102
On independent domination numbers of grid and toroidal grid directed graphs
Ramy
Shaheen
ٍSyrian
author
text
article
2019
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
4
v.
1
no.
2019
71
77
http://comb-opt.azaruniv.ac.ir/article_13846_0948ec1c34ebfc23d9e6b9f6dc3f735d.pdf
dx.doi.org/10.22049/cco.2019.26282.1090