Communications in Combinatorics and OptimizationCommunications in Combinatorics and Optimization
http://comb-opt.azaruniv.ac.ir/
Fri, 10 Apr 2020 23:04:49 +0100FeedCreatorCommunications in Combinatorics and Optimization
http://comb-opt.azaruniv.ac.ir/
Feed provided by Communications in Combinatorics and Optimization. Click to visit.On the super domination number of graphs
http://comb-opt.azaruniv.ac.ir/article_13980_0.html
The open neighborhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $Dsubseteq V(G)$, we define $overline{D}=V(G)setminus D$. A set $Dsubseteq V(G)$ is called a super dominating set of $G$ if for every vertex $uin overline{D}$, there exists $vin D$ such that $N(v)cap overline{D}={u}$. The super domination number of $G$ is the minimum cardinality among all super dominating sets of $G$. In this paper, we obtain closed formulas and tight bounds for the super domination number of $G$ in terms of several invariants of $G$. We also obtain results on the super domination number of corona product graphs and Cartesian product graphs.Sat, 26 Oct 2019 20:30:00 +0100Some new bounds on the general sum--connectivity index
http://comb-opt.azaruniv.ac.ir/article_13987_0.html
Let $G=(V,E)$ be a simple connectedgraph with $n$ vertices, $m$ edges and sequence of vertex degrees$d_1 ge d_2 ge cdots ge d_n>0$, $d_i=d(v_i)$, where $v_iin V$. With $isim j$ we denote adjacency ofvertices $v_i$ and $v_j$. The generalsum--connectivity index of graph is defined as $chi_{alpha}(G)=sum_{isim j}(d_i+d_j)^{alpha}$, where $alpha$ is an arbitrary realnumber. In this paper we determine relations between $chi_{alpha+beta}(G)$ and $chi_{alpha+beta-1}(G)$, where $alpha$ and $beta$ are arbitrary real numbers, and obtain new bounds for $chi_{alpha}(G)$. Also, by the appropriate choice of parameters $alpha$ and $beta$, we obtain a number of old/new inequalities for different vertex--degree--based topological indices.Thu, 07 Nov 2019 20:30:00 +0100Weak signed Roman domination in graphs
http://comb-opt.azaruniv.ac.ir/article_13989_0.html
A {em weak signed Roman dominating function} (WSRDF) of a graph $G$ with vertex set $V(G)$ is defined as afunction $f:V(G)rightarrow{-1,1,2}$ having the property that $sum_{xin N[v]}f(x)ge 1$ for each $vin V(G)$, where $N[v]$ is theclosed neighborhood of $v$. The weight of a WSRDF is the sum of its function values over all vertices.The weak signed Roman domination number of $G$, denoted by $gamma_{wsR}(G)$, is the minimum weight of a WSRDF in $G$.We initiate the study of the weak signed Roman domination number, and we present different sharp bounds on $gamma_{wsR}(G)$.In addition, we determine the weak signed Roman domination number of some classes of graphs.Tue, 12 Nov 2019 20:30:00 +0100New results on upper domatic number of graphs
http://comb-opt.azaruniv.ac.ir/article_13993_0.html
For a graph $G = (V, E)$, a partition $pi = {V_1,$ $V_2,$ $ldots,$ $V_k}$ of the vertex set $V$ is an textit{upper domatic partition} if $V_i$ dominates $V_j$ or $V_j$ dominates $V_i$ or both for every $V_i, V_j in pi$, whenever $i neq j$. The textit{upper domatic number} $D(G)$ is the maximum order of an upper domatic partition. We study the properties of upper domatic number and propose an upper bound in terms of clique number. Further, we discuss the upper domatic number of certain graph classes including unicyclic graphs and power graphs of paths and cycles.Wed, 11 Dec 2019 20:30:00 +0100Nonnegative signed total Roman domination in graphs
http://comb-opt.azaruniv.ac.ir/article_13992_0.html
‎Let $G$ be a finite and simple graph with vertex set $V(G)$‎. ‎A nonnegative signed total Roman dominating function (NNSTRDF) on a‎ ‎graph $G$ is a function $f:V(G)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin N(v)}f(x)ge 0$ for each‎ ‎$vin V(G)$‎, ‎where $N(v)$ is the open neighborhood of $v$‎, ‎and (ii) every vertex $u$ for which‎ ‎$f(u)=-1$ has a neighbor $v$ for which $f(v)=2$‎. ‎The weight of an NNSTRDF $f$ is $omega(f)=sum_{vin V (G)}f(v)$‎. ‎The nonnegative signed total Roman domination number $gamma^{NN}_{stR}(G)$‎ ‎of $G$ is the minimum weight of an NNSTRDF on $G$‎. ‎In this paper we‎‎initiate the study of the nonnegative signed total Roman domination number‎ ‎of graphs‎, ‎and we present different bounds on $gamma^{NN}_{stR}(G)$‎. ‎We determine the nonnegative signed total Roman domination‎‎number of some classes of graphs‎. ‎If $n$ is the order and $m$ the size‎‎of the graph $G$‎, ‎then we show that‎ ‎$gamma^{NN}_{stR}(G)ge frac{3}{4}(sqrt{8n+1}+1)-n$ and $gamma^{NN}_{stR}(G)ge (10n-12m)/5$‎. ‎In addition‎, ‎if $G$ is a bipartite graph of order $n$‎, ‎then we prove‎‎that $gamma^{NN}_{stR}(G)ge frac{3}{2}(sqrt{4n+1}-1)-n$‎.Mon, 09 Dec 2019 20:30:00 +0100Total Roman domination subdivision number in graphs
http://comb-opt.azaruniv.ac.ir/article_13997_0.html
A {em Roman dominating function} on a graph $G$ is a function $f:V(G)rightarrow {0,1,2}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. A {em total Roman dominating function} is a Roman dominating function with the additional property that the subgraph of $G$ induced by the set of all vertices of positive weight has no isolated vertices. The weight of a total Roman dominating function $f$ is the value $Sigma_{uin V(G)}f(u)$. The {em total Roman domination number} of $G$, $gamma_{tR}(G)$, is the minimum weight of a total Roman dominating function in $G$.The {em total Roman domination subdivision number} ${rmsd}_{gamma_{tR}}(G)$ of a graph $G$ is the minimum number of edges that must besubdivided (each edge in $G$ can be subdivided at most once) inorder to increase the total Roman domination number. In this paper,we initiate the study of total Roman domination subdivisionnumber in graphs and we present sharp bounds for this parameter.Sat, 11 Jan 2020 20:30:00 +0100On the Variance-Type Graph Irregularity Measures
http://comb-opt.azaruniv.ac.ir/article_14026_0.html
Bell's degree-variance Var$!{}_{B}$ for a graph $G$, with the degree sequence ($d_1,d_2,cdots,d_n$) and size $m$, is defined as$Var!_{B} (G)=frac{1}{n} sum _{i=1}^{n}left[d_{i} -frac{2m}{n}right]^{2}$.In this paper, a new version of the irregularity measures of variance-type, denoted by $Var_q$, is introduced and discussed. Based on a comparative study, it is demonstrated that the newly proposed irregularity measure $Var_q$ possess a better discrimination ability than the classical Bell's degree-variance in several cases.Wed, 01 Apr 2020 19:30:00 +0100Weak signed Roman k-domination in graphs
http://comb-opt.azaruniv.ac.ir/article_14019_0.html
Let $kge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$.A weak signed Roman $k$-dominating function (WSRkDF) on a graph $G$ is a function$f:V(G)rightarrow{-1,1,2}$ satisfying the conditions that $sum_{xin N[v]}f(x)ge k$ for eachvertex $vin V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRkDF $f$ is$w(f)=sum_{vin V(G)}f(v)$. The weak signed Roman $k$-domination number $gamma_{wsR}^k(G)$ of $G$ is theminimum weight of a WSRkDF on $G$. In this paper we initiate the study of the weak signed Roman $k$-dominationnumber of graphs, and we present different bounds on $gamma_{wsR}^k(G)$. In addition, we determine theweak signed Roman $k$-domination number of some classes of graphs. Some of our results are extensions ofwell-known properties of the signed Roman $k$-domination number $gamma_{sR}^k(G)$,introduced and investigated by Henning and Volkmann cite{hv16} as well as Ahangar, Henning, Zhao, L"{o}wenstein andSamodivkin cite{ahzls} for the case $k=1$.Thu, 12 Mar 2020 20:30:00 +0100Twin signed total Roman domatic numbers in digraphs
http://comb-opt.azaruniv.ac.ir/article_14024_0.html
Let $D$ be a finite simple digraph with vertex set $V(D)$ and arcset $A(D)$. A twin signed total Roman dominating function (TSTRDF) on thedigraph $D$ is a function $f:V(D)rightarrow{-1,1,2}$ satisfyingthe conditions that (i) $sum_{xin N^-(v)}f(x)ge 1$ and$sum_{xin N^+(v)}f(x)ge 1$ for each $vin V(D)$, where $N^-(v)$(resp. $N^+(v)$) consists of all in-neighbors (resp.out-neighbors) of $v$, and (ii) every vertex $u$ for which$f(u)=-1$ has an in-neighbor $v$ and an out-neighbor $w$ with$f(v)=f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinct twin signed total Romandominating functions on $D$ with the property that $sum_{i=1}^df_i(v)le 1$for each $vin V(D)$, is called a twin signed total Roman dominating family (offunctions) on $D$. The maximum number of functions in a twin signed total Romandominating family on $D$ is the twin signed total Roman domatic number of $D$,denoted by $d_{stR}^*(D)$. In this paper, we initiate the study of the twinsigned total Roman domatic number in digraphs and we present some sharp bounds on$d_{stR}^*(D)$. In addition, we determine the twin signed total Roman domatic numberof some classes of digraphs.Sat, 14 Mar 2020 20:30:00 +0100On relation between the Kirchhoff index and number of spanning trees of graph
http://comb-opt.azaruniv.ac.ir/article_13873_2294.html
Let $G=(V,E)$, $V={1,2,ldots,n}$, $E={e_1,e_2,ldots,e_m}$,be a simple connected graph, with sequence of vertex degrees$Delta =d_1geq d_2geqcdotsgeq d_n=delta >0$ and Laplacian eigenvalues$mu_1geq mu_2geqcdotsgeqmu_{n-1}>mu_n=0$. Denote by $Kf(G)=nsum_{i=1}^{n-1}frac{1}{mu_i}$ and $t=t(G)=frac 1n prod_{i=1}^{n-1} mu_i$ the Kirchhoff index and number of spanning trees of $G$, respectively. In this paper we determine several lower bounds for $Kf(G)$ depending on $t(G)$ and some of the graph parameters $n$, $m$, or $Delta$.Sun, 31 May 2020 19:30:00 +0100A study on some properties of leap graphs
http://comb-opt.azaruniv.ac.ir/article_13876_2294.html
In a graph G, the first and second degrees of a vertex v is equal to thenumber of their first and second neighbors and are denoted by d(v/G) andd 2 (v/G), respectively. The first, second and third leap Zagreb indices are thesum of squares of second degrees of vertices of G, the sum of products of second degrees of pairs of adjacent vertices in G and the sum of products of firstand second degrees of vertices of G, respectively. In this paper, we initiate in studying a new class of graphs depending on the relationship between firstand second degrees of vertices and is so-called a leap graph. Some propertiesof the leap graphs are presented. All leap trees and {C 3, C 4 }-free leap graphsare characterized.Sun, 31 May 2020 19:30:00 +0100A note on the Roman domatic number of a digraph
http://comb-opt.azaruniv.ac.ir/article_13884_2294.html
Roman dominating function} on a digraph $D$ with vertex set $V(D)$ is a labeling$fcolon V(D)to {0, 1, 2}$such that every vertex with label $0$ has an in-neighbor with label $2$. A set ${f_1,f_2,ldots,f_d}$ ofRoman dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vin V(D)$,is called a {em Roman dominating family} (of functions) on $D$. The maximum number of functions in aRoman dominating family on $D$ is the {em Roman domatic number} of $D$, denoted by $d_{R}(D)$.In this note, we study the Roman domatic number in digraphs, and we present some sharpbounds for $d_{R}(D)$. In addition, we determine the Roman domatic number of some digraphs.Some of our results are extensions of well-known properties of the Roman domatic number ofundirected graphs.Sun, 31 May 2020 19:30:00 +0100Total double Roman domination in graphs
http://comb-opt.azaruniv.ac.ir/article_13945_2294.html
Let $G$ be a simple graph with vertex set $V$. A double Roman dominating function (DRDF) on $G$ is a function $f:Vrightarrow{0,1,2,3}$ satisfying that if $f(v)=0$, then the vertex $v$ must be adjacent to at least two vertices assigned $2$ or one vertex assigned $3$ under $f$, whereas if $f(v)=1$, then the vertex $v$ must be adjacent to at least one vertex assigned $2$ or $3$. The weight of a DRDF $f$ is the sum $sum_{vin V}f(v)$. A total double Roman dominating function (TDRDF) on a graph $G$ with no isolated vertex is a DRDF $f$ on $G$ with the additional property that the subgraph of $G$ induced by the set ${vin V:f(v)ne0}$ has no isolated vertices. The total double Roman domination number $gamma_{tdR}(G)$ is the minimum weight of a TDRDF on $G$. In this paper, we give several relations between the total double Roman domination number of a graph and other domination parameters and we determine the total double Roman domination number of some classes of graphs.Sun, 31 May 2020 19:30:00 +0100On the edge geodetic and edge geodetic domination numbers of a graph
http://comb-opt.azaruniv.ac.ir/article_13946_2294.html
In this paper, we study both concepts of geodetic dominatingand edge geodetic dominating sets and derive some tight upper bounds onthe edge geodetic and the edge geodetic domination numbers. We also obtainattainable upper bounds on the maximum number of elements in a partitionof a vertex set of a connected graph into geodetic sets, edge geodetic sets,geodetic dominating sets and edge geodetic dominating sets, respectively.Sun, 31 May 2020 19:30:00 +0100The topological ordering of covering nodes
http://comb-opt.azaruniv.ac.ir/article_13958_2294.html
The topological ordering algorithm sorts nodes of a directed graph such that the order of the tail of each arc is lower than the order of its head. In this paper, we introduce the notion of covering between nodes of a directed graph. Then, we apply the topological orderingalgorithm on graphs containing the covering nodes. We show that there exists a cut set withforward arcs in these graphs and the order of the covering nodes is successive.Sun, 31 May 2020 19:30:00 +0100Characterization of signed paths and cycles admitting minus dominating function
http://comb-opt.azaruniv.ac.ir/article_13977_2294.html
If G = (V, E, σ) is a finite signed graph, a function f : V → {−1, 0, 1} is a minusdominating function (MDF) of G if f(u) +summation over all vertices v∈N(u) of σ(uv)f(v) ≥ 1 for all u ∈ V . In this paper we characterize signed paths and cycles admitting an MDF.Sun, 31 May 2020 19:30:00 +0100The 2-dimension of a Tree
http://comb-opt.azaruniv.ac.ir/article_13979_2294.html
Let $x$ and $y$ be two distinct vertices in a connected graph $G$. The $x,y$-location of a vertex $w$ is the ordered pair of distances from $w$ to $x$ and $y$, that is, the ordered pair $(d(x,w), d(y,w))$. A set of vertices $W$ in $G$ is $x,y$-located if any two vertices in $W$ have distinct $x,y$-location.A set $W$ of vertices in $G$ is 2-located if it is $x,y$-located, for some distinct vertices $x$ and $y$. The 2-dimension of $G$ is the order of a largest set that is 2-located in $G$. Note that this notion is related to the metric dimension of a graph, but not identical to it.We study in depth the trees $T$ that have a 2-locating set, that is, have 2-dimension equal to the order of $T$. Using these results, we have a nice characterization of the 2-dimension of arbitrary trees.Sun, 31 May 2020 19:30:00 +0100