A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G-D$ has a~neighbor in $D$, while $D$ is a paired-dominating set of $G$ if $D$ is a~dominating set and the subgraph induced by $D$ contains a perfect matching. A graph $G$ is a $D\!P\!D\!P$-graph if it has a pair $(D,P)$ of disjoint sets of vertices of $G$ such that $D$ is a dominating set and $P$ is a paired-dominating set of $G$. The study of the $D\!P\!D\!P$-graphs was initiated by Southey and Henning (Cent. Eur. J. Math. 8 (2010) 459--467; J. Comb. Optim. 22 (2011) 217--234). In this paper, we provide conditions which ensure that a graph is a $D\!P\!D\!P$-graph. In particular, we characterize the minimal $D\!P\!D\!P$-graphs.
A graph $G$ is a $D\!D_2$-graph if it has a pair $(D,D_2)$ of disjoint sets of vertices of $G$ such that $D$ is a dominating set and $D_2$ is a 2-dominating set of $G$. We provide several characterizations and hardness results concerning $D\!D_2$-graphs.
A graph $G$ is a DTDP-graph if it has a pair $(D,T)$ of disjoint sets of vertices of $G$ such that $D$ is a dominating set and $T$ is a total dominating set of $G$. Such graphs were studied in a number of research papers. In this paper we study further properties of DTDP-graphs and, in particular, we characterize minimal DTDP-gaphs without loops.
Let $G=(V,E)$ be a simple graph without isolated vertices. A set $S\subseteq V$ is a paired-dominating set if every vertex in $V-S$ has at least one neighbor in $S$ and the subgraph induced by $S$ contains a perfect matching. In this paper, we present a linear-time algorithm to determine whether a given vertex in a block graph is contained in all its minimum paired-dominating sets.
A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G\setminus D$ has a neighbor in $D$, while $D$ is a 2-dominating set of $G$ if every vertex belonging to $V_G\setminus D$ is joined by at least two edges with a vertex or vertices in $D$. A graph $G$ is a $(2,2)$-dominated graph if it has a pair $(D,D')$ of disjoint $2$-dominating sets of vertices of $G$. In this paper we present two characterizations of minimal $(2,2)$-dominated graphs.
Let $G=(V,E)$ be a graph without isolated vertices. A matching in $G$ is a set of independent edges in $G$. A perfect matching $M$ in $G$ is a matching such that every vertex of $G$ is incident to an edge of $M$. A set $S\subseteq V$ is a \textit{paired-dominating set} of $G$ if every vertex in $V-S$ is adjacent to some vertex in $S$ and if the subgraph $G[S]$ induced by $S$ contains at least one perfect matching. The paired-domination problem is to find a paired-dominating set of $G$ with minimum cardinality. A set $MPD\subseteq E$ is a \textit{matched-paired-dominating set} of $G$ if $MPD$ i…
In this paper, we continue the study of locating-paired-dominating set, abbreviated LPDS, in graphs introduced by McCoy and Henning. Given a finite or infinite graph $G=(V,E)$, a set $S\subset V$ is paired-dominating if the induced subgraph $G[S]$ has a perfect matching and every vertex in $V$ is adjacent to a vertex in $S$. The other condition for LPDS requires that for any distinct vertices $u,v \in V\backslash S$, we have $N(u)\cap S\neq N(v)\cap S$. Motivated by the conjecture of Kinawi, Hussain and Niepel, we prove the minimal density of LPDS in the king grid is between $8/37$ and $2/9$, …
Let G be a simple undirected graph with no isolated vertex. A paired dominating set of G is a dominating set which induces a subgraph that has a perfect matching. The paired domination number of G, denoted by γpr(G), is the size of its smallest paired dominating set. Goddard and Henning conjectured that γpr(G) {\leq} 4n/7 holds for every graph G with δ(G) {\geq} 3, except the Petersen Graph. In this paper, we prove this conjecture for cubic graphs.
In this paper, we consider dominating sets $D$ and $D'$ such that $D$ and $D'$ are disjoint and there exists a perfect matching between them. Let $DD_{\textrm{m}}(G)$ denote the cardinality of smallest such sets $D, D'$ in $G$ (provided they exist, otherwise $DD_{\textrm{m}}(G) = \infty$). This concept was introduced in [Klostermeyer et al., Theory and Application of Graphs, 2017] in the context of studying a certain graph protection problem. We characterize the trees $T$ for which $DD_{\textrm{m}}(T)$ equals a certain graph protection parameter and for which $DD_{\textrm{m}}(T) = α(T)$, where…
We show that every simple planar near-triangulation with minimum degree at least three contains two disjoint total dominating sets. The class includes all simple planar triangulations other than the triangle. This affirms a conjecture of Goddard and Henning [Thoroughly dispersed colorings, J. Graph Theory, 88 (2018) 174-191].
The balanced hypercube $BH_{n}$, proposed by Wu and Huang, is a variation of the hypercube. The paired 1-disjoint path cover of $BH_{n}$ is the Hamiltonian laceability, which was obtained by Xu et al. in [Appl. Math. Comput. 189 (2007) 1393--1401]. The paired 2-disjoint path cover of $BH_{n}$ was obtained by Cheng et al. in [Appl. Math. and Comput. 242 (2014) 127-142]. In this paper, we obtain the paired 3-disjoint path cover of $BH_{n}$ with $n\geq 3$. This result improves the above known results about the paired $k$-disjoint path covers of $BH_{n}$ for $k=1,2$.
Paired 2-disjoint path covers of Bcube under the partitioned edge fault model
College of Computer Science, Nankai University（南开大学计算机学院）；Academy for Advanced Interdisciplinary Studies, Nankai University（南开大学前沿交叉科学研究院）；Institute of Information and Decision Sciences, National Taipei University of Business（台北商业大学信息与决策科学研究所）
Let $G = (V, E)$ be a graph without an isolated vertex. A set $D\subseteq V(G)$ is a $k$-distance paired domination set of $G$ if $D$ is a $k$-distance dominating set of $G$ and the induced subgraph $\langle D \rangle$ has a perfect matching. The minimum cardinality of a $k$-distance paired dominating set for graph $G$ is the $k$-distance paired domination number, denoted by $γ_{p} ^{k}(G)$. In this paper, the $k$-distance paired domination of the flower graph $f_{n\times m}$ is discussed. For $m,n\geq 3$, the exact values for paired domination number and $2$-distance paired domination number …
As a variant of the well-known hypercube, the balanced hypercube $BH_n$ was proposed as a novel interconnection network topology for parallel computing. It is known that $BH_n$ is bipartite. Assume that $S=\{s_1,s_2\}$ and $T=\{t_1,t_2\}$ are any two sets of two vertices in different partite sets of $BH_n$ ($n\geq1$). It has been proved that there exist two vertex-disjoint $s_1,t_1$-path and $s_2,t_2$-path of $BH_n$ covering all vertices of it. In this paper, we prove that there always exist two vertex-disjoint $s_1,t_1$-path and $s_2,t_2$-path covering all vertices of $BH_n$ with at most $2n-…
A set $D$ of vertices in a graph $G$ is called dominating if every vertex of $G$ is either in $D$ or adjacent to a vertex of $D$. The paired domination number $γ_{\mathrm{pr}}(G)$ of $G$ is the minimum size of a dominating set whose induced subgraph admits a perfect matching, and the upper domination number $Γ(G)$ is the maximum size of a minimal dominating set. In this paper, we investigate the sharpness of two multiplicative inequalities for these domination parameters, where the graph product is the direct product $\times$. We show that for every positive constant $c$, there exist graphs $G…
Let $G=(V,E)$ be a graph without isolated vertices. A set $S\subseteq V$ is a paired-domination set if every vertex in $V-S$ is adjacent to a vertex in $S$ and the subgraph induced by $S$ contains a perfect matching. The paired-domination problem is to determine the paired-domination number, which is the minimum cardinality of a paired-dominating set. Motivated by a mistaken algorithm given by Chen, Kang and Ng [ Paired domination on interval and circular-arc graphs, Disc. Appl. Math. 155(2007),2077-2086], we present two linear time algorithms to find a minimum cardinality paired-dominating se…
Let $G$ be a graph without isolated vertices. The total domination number of $G$ is the minimum number of vertices that can dominate all vertices in $G$, and the paired domination number of $G$ is the minimum number of vertices in a dominating set whose induced subgraph contains a perfect matching. This paper determines the total domination number and the paired domination number of the toroidal meshes, i.e., the Cartesian product of two cycles $C_n$ and $C_m$ for any $n\ge 3$ and $m\in\{3,4\}$, and gives some upper bounds for $n, m\ge 5$.
A locating-dominating set of a graph $G$ is a dominating set of $G$ such that every vertex of $G$ outside the dominating set is uniquely identified by its neighborhood within the dominating set. The location-domination number of $G$ is the minimum cardinality of a locating-dominating set in $G$. Let $G_{1}$ and $G_{2}$ be the disjoint copies of a graph $G$ and $f:V(G_{1})\rightarrow V(G_{2})$ be a function. A functigraph $F^f_{G}$ consists of the vertex set $V(G_{1})\cup V(G_{2})$ and the edge set $E(G_{1})\cup E(G_{2})\cup \{uv:v=f(u)\}$. In this paper, we study the variation of the location-…
We introduce and study a Maker-Breaker type game in which the issue is to create or avoid two disjoint dominating sets in graphs without isolated vertices. We prove that the maker has a winning strategy on all connected graphs if the game is started by the breaker. This implies the same in the $(2:1)$ biased game also in the maker-start game. It remains open to characterize the maker-win graphs in the maker-start non-biased game, and to analyze the $(a:b)$ biased game for $(a:b)\neq (2:1)$. For a more restricted variant of the non-biased game we prove that the maker can win on every graph with…
A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $γ(G)$ is the size of a minimum dominating set in $G$. Vizing's conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $γ(G) γ(H) \leq γ(G \Box H)$, and Clark and Suen (2000) proved that $γ(G) γ(H) \leq 2γ(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ t…
The coalition in a graph $G$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a dominating set but whose union $V_{1}\cup V_{2}$, is a dominating set. A coalition partition in a graph $G$ is a vertex partition $π$ = $\{V_1, V_2,..., V_k \}$ such that every set $V_i \in π$ is not a dominating set but forms a coalition with another set $V_j\in π$ which is not a dominating set. The coalition number $C(G)$ equals the maximum $k$ of a coalition partition of $G$. In this paper, we compute the coalition number of all cubic graphs of order at most $10$.
Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $γ(G)$, is the domination number of $G$. A dominating set $S$ is called a super dominating set of $G$, if for every vertex $u\in \overline{S}=V-S$, there exists $v\in S$ such that $N(v)\cap \overline{S}=\{u\}$. The cardinality of a smallest super dominating set of $G$, denoted by $γ_{sp}(G)$, is the super domination number of $G$. In this paper, we obtain more results on t…
Given a graph $G=(V(G), E(G))$, the size of a minimum dominating set, minimum paired dominating set, and a minimum total dominating set of a graph $G$ are denoted by $γ(G)$, $γ_{\rm pr}(G)$, and $γ_{t}(G)$, respectively. For a positive integer $k$, a $k$-packing in $G$ is a set $S \subseteq V(G)$ such that for every pair of distinct vertices $u$ and $v$ in $S$, the distance between $u$ and $v$ is at least $k+1$. The $k$-packing number is the order of a largest $k$-packing and is denoted by $ρ_{k}(G)$. It is well known that $γ_{\rm pr}(G) \le 2γ(G)$. In this paper, we prove that it is NP-hard t…
Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meet these bounds.
Given a digraph $D=(V,A)$, a set $B\subset V$ is a packing set in $D$ if there are no arcs joining vertices of $B$ and for any two vertices $x,y\in B$ the sets of in-neighbors of $x$ and $y$ are disjoint. The set $S$ is a dominating set (an open dominating set) in $D$ if every vertex not in $S$ (in $V$) has an in-neighbor in $S$. Moreover, a dominating set $S$ is called a total dominating set if the subgraph induced by $S$ has no isolated vertices. The packing sets of maximum cardinality and the (total, open) dominating sets of minimum cardinality in digraphs are studied in this article. We pr…
For a graph $G=(V,E)$, a subset $D$ of vertex set $V$, is a dominating set of $G$ if every vertex not in $D$ is adjacent to atleast one vertex of $D$. A dominating set $D$ of a graph $G$ with no isolated vertices is called a paired dominating set (PD-set), if $G[D]$, the subgraph induced by $D$ in $G$ has a perfect matching. The Min-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the Min-PD problem remains NP-complete even when $G$ belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the positive side, the problem is effic…
Let $G$ be a graph with vertex set $V$. Two disjoint sets $V_1, V_2\subseteq V$ are called a total coalition in $G$, if neither $V_1$ and $V_2$ is a total dominating set of $G$ but $V_1\cup V_2$ is a total dominating set. A total coalition partition of $G$ is a vertex partition $π=\{V_1,V_2,\ldots, V_k\}$ such that no set of $π$ is a total dominating set but each set $V_i\in π$ forms a total coalition with another set $V_j\in π$. The maximum cardinality of a total coalition partition of $G$ is called the total coalition number of $G$, denoted by $TC(G)$. In this paper, we initiate the study of…
For a graph $G$, the $γ$-graph of $G$, $G(γ)$, is the graph whose vertices correspond to the minimum dominating sets of $G$, and where two vertices of $G(γ)$ are adjacent if and only if their corresponding dominating sets in $G$ differ by exactly two adjacent vertices. In this paper, we present several variations of the $γ$-graph including those using identifying codes, locating-domination, total-domination, paired-domination, and the upper-domination number. For each, we show that for any graph $H$, there exist infinitely many graphs whose $γ$-graph variant is isomorphic to $H$.
University College London（伦敦大学学院）；Simon Fraser University（西蒙菲莎大学）；University of Ljubljana（卢布尔雅那大学）；McGill University（麦吉尔大学）；ETH Zürich（苏黎世联邦理工学院）；University of Oxford（牛津大学）；Monash University（蒙纳士大学）
The connected coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a connected dominating set but whose union $V_{1}\cup V_{2}$, is a connected dominating set. A connected coalition partition in a graph $G$ of order $n=|V|$ is a vertex partition $ψ$ = $\{V_1, V_2,..., V_k \}$ such that every set $V_i \in ψ$ either is a connected dominating set consisting of a single vertex of degree $n-1$, or is not a connected dominating set but forms a connected coalition with another set $V_j\in ψ$ which is not a connected dominating set. The conn…
Let $F_1, F_2, ..., F_k$ be graphs with the same vertex set $V$. A subset $S \subseteq V$ is a simultaneous dominating set if for every $i$, $1 \le i \le k$, every vertex of $F_i$ not in $S$ is adjacent to a vertex in $S$ in $F_i$; that is, the set $S$ is simultaneously a dominating set in each graph $F_i$. The cardinality of a smallest such set is the simultaneous domination number. We present general upper bounds on the simultaneous domination number. We investigate bounds in special cases, including the cases when the factors, $F_i$, are $r$-regular or the disjoint union of copies of $K_r$.…
Let $G(V,E)$ be a finite, simple, isolate-free graph. Two disjoint sets $A,B\subset V$ form a total coalition in $G$, if none of them is a total dominating set, but their union $A\cup B$ is a total dominating set. A vertex partition $Ψ=\{C_1,C_2,\dots,C_k\}$ is a total coalition partition, if none of the partition classes is a total dominating set, meanwhile for every $i\in\{1,2,\dots,k\}$ there exists a distinct $j\in\{1,2,\dots,k\}$ such that $C_i$ and $C_j$ form a total coalition. The maximum cardinality of a total coalition partition of $G$ is the total coalition number of $G$ and denoted …
Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and denoted by $γ_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum d_t(G,i)$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and ident…
The balanced hypercube $BH_n$, a variant of the hypercube, was proposed as a desired interconnection network topology. It is known that $BH_n$ is bipartite. Assume that $S=\{s_1,s_2,\cdots,s_{2n-2}\}$ and $T=\{t_1,t_2,\cdots,t_{2n-2}\}$ are any two sets of vertices in different partite sets of $BH_n$ ($n\geq2$). It has been proved that there exists paired 2-disjoint path cover of $BH_n$. In this paper, we prove that there exists unpaired $(2n-2)$-disjoint path cover of $BH_n$ ($n\geq2$) from $S$ to $T$, which improved some known results. The upper bound $2n-2$ of the number of disjoint paths i…
A matching in a hypergraph $H$ is a set of pairwise disjoint hyperedges. The matching number $α'(H)$ of $H$ is the size of a maximum matching in $H$. A subset $D$ of vertices of $H$ is a dominating set of $H$ if for every $v\in V\setminus D$ there exists $u\in D$ such that $u$ and $v$ lie in an hyperedge of $H$. The cardinality of a minimum dominating set of $H$ is called the domination number of $H$, denoted by $γ(H)$. It is known that for a intersecting hypergraph $H$ with rank $r$, $γ(H)\leq r-1$. In this paper we present structural properties on intersecting hypergraphs with rank $r$ satis…
