Paired-dominating Set Research Articles

Leaf sector covers with applications on circle graphs

Damian and Pemmaraju (2002) [8] introduced leaf sector covers for circle graphs and presented an O(n2)-time algorithm to find this data structure. They subsequently employed it as an algorithmic tool, leading to the development of an 8-approximation algorithm for the dominating set problem, a (2+ε)-approximation scheme for the dominating set problem, and a (3+ε)-approximation scheme for the total dominating set problem.In this paper, we have improved the time complexity from O(n2) to O(n+m) for finding a leaf sector cover. Furthermore, we employed leaf sector covers as a powerful algorithmic tool to design a 10-approximation algorithm for the total dominating set problem and a 12-approximation algorithm for the paired-dominating set problem. Moreover, we also proposed a (2+ε)-approximation scheme for the total dominating set problem and a (4+ε)-approximation scheme for the paired-dominating set problem. The results presented above are currently the best algorithms for circle graphs.

Cubic graphs have paired-domination number at most four-seventh of their orders

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number γpr(G) is the minimum cardinality of a paired-dominating set of G. In this paper, we prove the conjecture of Goddard and Henning (2009) stating that if G is a connected cubic graph of order n and G is different from the Petersen graph, then γpr(G)≤4n/7.

On the Paired-Domination Subdivision Number of Trees

A paired-dominating set of a graph G without isolated vertices is a dominating set of vertices whose induced subgraph has perfect matching. The minimum cardinality of a paired-dominating set of G is called the paired-domination number γpr(G) of G. The paired-domination subdivision number sdγpr(G) of G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the paired-domination number. Here, we show that, for each tree T ≠ P5 of order n ≥ 3 and each edge e ∉ E(T), sdγpr(T) + sdγpr(T + e) ≤ n + 2.

Minimal graphs with disjoint dominating and paired-dominating sets

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.

Paired-Domination Problem on Distance-Hereditary Graphs

A paired-dominating set of a graph G is a dominating set S of G such that the subgraph of G induced by S has a perfect matching. Haynes and Slater (Networks 32(3):199–206, 1998) introduced the concept of paired-domination and showed that the problem of determining minimum paired-dominating sets is NP-complete on general graphs. Ever since then many algorithmic results are studied on some important classes of graphs. In this paper, we extend the results by providing an $$O(n^2)$$ -time algorithm on distance-hereditary graphs.

Algorithmic aspects of upper paired-domination in graphs

A set D of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to a vertex in D and the subgraph induced by D contains a perfect matching (not necessarily as an induced subgraph). A paired-dominating set of G is minimal if no proper subset of it is a paired-dominating set of G. The upper paired-domination number of G, denoted by Γpr(G), is the maximum cardinality of a minimal paired-dominating set of G. In Upper-PDS, it is required to compute a minimal paired-dominating set with cardinality Γpr(G) for a given graph G. In this paper, we show that Upper-PDS cannot be approximated within a factor of n1−ε for any ε>0, unless P=NP and Upper-PDS is APX-complete for bipartite graphs of maximum degree 4. On the positive side, we show that Upper-PDS can be approximated within O(Δ)-factor for graphs with maximum degree Δ. We also show that Upper-PDS is solvable in polynomial time for threshold graphs, chain graphs, and proper interval graphs.

Disjoint Paired-Dominating sets in Cubic Graphs

A paired-dominating set of a graph G is a dominating set D with the additional requirement that the induced subgraph G[D] contains a perfect matching. We prove that the vertex set of every claw-free cubic graph can be partitioned into two paired-dominating sets.

Paired-domination in claw-free graphs with minimum degree at least three

Let G=(V,E) be a simple graph without isolated vertices. A set S⊆V is called 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. The paired-domination number of G, denoted by γpr(G), is the minimum cardinality of a paired-dominating set of G. We show that γpr(G)≤4n∕7 if G is a claw-free graph of order n with minimum degree at least three. The statement partly confirms the conjecture proposed by Goddard and Henning in 2009.

Computing a minimum paired-dominating set in strongly orderable graphs

A set D⊆V of a graph G=(V,E) is called a dominating set of G if every vertex in V∖D has at least one neighbor in D. A dominating set D of G is a paired-dominating set of G if G[D], the subgraph of G induced by D, has a perfect matching. Given a graph G, Min-Paired-Dom-Set is the problem to find a paired-dominating set of G of minimum cardinality. Min-Paired-Dom-Set is known to be NP-hard for general graphs and many other restricted classes of graphs. However, Min-Paired-Dom-Set is solvable in polynomial time in some graph classes including strongly chordal graphs and chordal bipartite graphs. In this paper, we strengthen this result by proposing a polynomial time algorithm to compute a minimum paired-dominating set in the class of strongly orderable graphs, which includes strongly chordal graphs and chordal bipartite graphs. The algorithm runs in linear time if a quasi-simple elimination ordering of the strongly orderable graph is provided.

Total domination versus paired-domination in regular graphs

A subset S of vertices of a graph G is a dominating set of G if every vertex not in S has a neighbor in S, while S is a total dominating set of G if every vertex has a neighbor in S. If S is a dominating set with the additional property that the subgraph induced by S contains a perfect matching, then S is a paired-dominating set. The domination number, denoted γ(G), is the minimum cardinality of a dominating set of G, while the minimum cardinalities of a total dominating set and paired-dominating set are the total domination number, \gt(G), and the paired-domination number, \gp(G), respectively. For k ≥ 2, let G be a connected k-regular graph. It is known [Schaudt, Total domination versus paired domination, Discuss. Math. Graph Theory 32 (2012) 435--447] that \gpr(G)/γt(G) \le (2 k)/(k + 1). In the special case when k = 2, we observe that \gpr(G)/γt(G) \le 4/3, with equality if and only if G \cong C5. When k = 3, we show that \gpr(G)/γt(G) \le 3/2, with equality if and only if G is the Petersen graph. More generally for k ≥ 2, if G has girth at least 5 and satisfies \gpr(G)/γt(G) = (2 k)/(k + 1), then we show that G is a diameter-2 Moore graph. As a consequence of this result, we prove that for k ≥ 2 and k \ne 57, if G has girth at least 5, then \gpr(G)/γt(G) \le (2 k)/(k + 1), with equality if and only if k=2 and G \cong C5 or k = 3 and G is the Petersen graph.

Paired-domination number of claw-free odd-regular graphs

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph, denoted by $$\gamma _{pr}(G)$$źpr(G). Let G be a connected $$\{K_{1,3}, K_{4}-e\}$${K1,3,K4-e}-free cubic graph of order n. We show that $$\gamma _{pr}(G)\le \frac{10n+6}{27}$$źpr(G)≤10n+627 if G is $$C_{4}$$C4-free and that $$\gamma _{pr}(G)\le \frac{n}{3}+\frac{n+6}{9(\lceil \frac{3}{4}(g_o+1)\rceil +1)}$$źpr(G)≤n3+n+69(ź34(go+1)ź+1) if G is $$\{C_{4}, C_{6}, C_{10}, \ldots , C_{2g_o}\}$${C4,C6,C10,ź,C2go}-free for an odd integer $$g_o\ge 3$$goź3; the extremal graphs are characterized; we also show that if G is a 2 -connected, $$\gamma _{pr}(G) = \frac{n}{3} $$źpr(G)=n3. Furthermore, if G is a connected $$(2k+1)$$(2k+1)-regular $$\{K_{1,3}, K_4-e\}$${K1,3,K4-e}-free graph of order n, then $$\gamma _{pr}(G)\le \frac{n}{k+1} $$źpr(G)≤nk+1, with equality if and only if $$G=L(F)$$G=L(F), where $$F\cong K_{1, 2k+2}$$FźK1,2k+2, or k is even and $$F\cong K_{k+1,k+2}$$FźKk+1,k+2.

Upper Bounds for the Paired-Domination Numbers of Graphs

A set $$S\subseteq V$$S⊆V is a paired-dominating set if every vertex in $$V{\setminus } S$$V\S has at least one neighbor in S and the subgraph induced by S contains a perfect matching. The paired-domination number of a graph G, denoted by $$\gamma _{pr}(G)$$?pr(G), is the minimum cardinality of a paired-dominating set of G. A conjecture of Goddard and Henning says that if G is not the Petersen graph and is a connected graph of order n with minimum degree $$\delta (G)\ge 3$$?(G)?3, then $$\gamma _{pr}(G)\le 4n/7$$?pr(G)≤4n/7. In this paper, we confirm this conjecture for k-regular graphs with $$k\ge 4$$k?4.

A linear-time algorithm for paired-domination on circular-arc graphs

In a graph G, a vertex subset S⊆V(G) is said to be a dominating set of G if every vertex not in S is adjacent to a vertex in S. A dominating set S of a graph G is called a paired-dominating set if the induced subgraph G[S] contains a perfect matching. The paired-domination problem involves finding a minimum paired-dominating set of G. For this problem, Chen et al. [J. Comb. Optim. 19 (4) (2010) 457–470] proposed an O(n+m)-time algorithm on interval graphs and Cheng et al. [Discrete Appl. Math. 155 (16) (2007) 2077–2086] designed an O(m(n+m))-time algorithm on circular-arc graphs. In this paper, we strengthen the results of Cheng et al. by showing an O(n+m)-time algorithm. Moreover, the algorithm can be completed in O(n) time if an intersection model of a circular-arc graph G with sorted endpoints is given. Since interval graphs are circular-arc graphs, we also obtain a linear time algorithm on interval graphs.

Domination Parameters in Coronene Torus Network

A set D of vertices in a graph G is said to be a dominating set if every vertex in V − D is adjacent to some vertex in D. A subset D of V is a paired-dominating set of G, if D is a dominating set and the subgraph induced by the vertices of D contains a perfect matching. A set D of a vertices in a graph G = (V, E) is said to be a total dominating set of G if every vertex in V is adjacent to some vertex in D. The total bondage number bt(G) of G is the minimum number of edges whose removal enlarges the total domination number. In this paper we determine the domination number, paired-domination number, bondage number and the total bondage number for coronene torus using H-packing.

The paired-domination and the upper paired-domination numbers of graphs

In this paper we continue the study of paired-domination in graphs. A paired-dominating set, abbreviated PDS, of a graph \(G\) with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of \(G\), denoted by \(\gamma_{p}(G)\), is the minimum cardinality of a PDS of \(G\). The upper paired-domination number of \(G\), denoted by \(\Gamma_{p}(G)\), is the maximum cardinality of a minimal PDS of \(G\). Let \(G\) be a connected graph of order \(n\geq 3\). Haynes and Slater in [Paired-domination in graphs, Networks 32 (1998), 199-206], showed that \(\gamma_{p}(G)\leq n-1\) and they determine the extremal graphs \(G\) achieving this bound. In this paper we obtain analogous results for \(\Gamma_{p}(G)\). Dorbec, Henning and McCoy in [Upper total domination versus upper paired-domination, Questiones Mathematicae 30 (2007), 1-12] determine \(\Gamma_{p}(P_n)\), instead in this paper we determine \(\Gamma_{p}(C_n)\). Moreover, we describe some families of graphs \(G\) for which the equality \(\gamma_{p}(G)=\Gamma_{p}(G)\) holds.

Locating–paired-dominating sets in square grids

A set S of vertices of a graph G is paired-dominating if S induces a matching in G and S dominates all vertices of G. A set S⊂V(G) is locating if for any two distinct vertices u,v∈V(G)∖S, N(u)∩S≠N(v)∩S, where N(u) and N(v) are open neighborhoods of vertices u and v. We give a complete characterization of locating–paired-dominating sets with minimal density in the infinite square grid Z2.

Graphs with maximum size and given paired-domination number

A paired-dominating set of a graph is a dominating set whose induced subgraph contains a perfect matching. The paired-domination number of a graph G is the minimum cardinality of a paired-dominating set in G. We determine the maximum possible number of edges in a graph with given order and given paired-domination number and we completely characterize the infinite family of graphs that achieve this maximum possible size. Our result builds on a classic result in 1962 due to Erdős and Rényi (1962) since the case when the paired-domination is four is equivalent to determining the minimum size of a nontrivial diameter-2 graph (excluding a star) in the complement of the graphs we are considering. More precisely, for k≥2, let G be a graph with paired-domination number 2k, order n≥2k, and size m. As a consequence of the Erdős–Rényi result it follows that if k=2, then m≤n2−k(n−2)+1. For k≥3, we show that m≤n2−k(n−2) and we characterize the graphs that achieve equality in this bound.

Bounding the paired-domination number of a tree in terms of its annihilation number

A paired-dominating set of a graph G=(V, E) with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by ?pr(G), is the minimum cardinality of a paired-dominating set of G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we prove that for any tree T of order n?2,?pr(T)? 4a(T)+2/3 and we characterize the trees achieving this bound.

2-Distance paired-dominating number of graphs

Let $$G=(V,E)$$ be a simple graph without isolated vertices. For a positive integer $$k$$ , a subset $$D$$ of $$V(G)$$ is a $$k$$ -distance paired-dominating set if each vertex in $$V\setminus {D}$$ is within distance $$k$$ of a vertex in $$D$$ and the subgraph induced by $$D$$ contains a perfect matching. In this paper, we give some upper bounds on the 2-distance paired-dominating number in terms of the minimum and maximum degree, girth, and order.

All graphs with paired-domination number two less than their order

Let \(G=(V,E)\) be a graph with no isolated vertices. A set \(S\subseteq V\) is a paired-dominating set of \(G\) if every vertex not in \(S\) is adjacent with some vertex in \(S\) and the subgraph induced by \(S\) contains a perfect matching. The paired-domination number \(\gamma_{p}(G)\) of \(G\) is defined to be the minimum cardinality of a paired-dominating set of \(G\). Let \(G\) be a graph of order \(n\). In [Paired-domination in graphs, Networks 32 (1998), 199-206] Haynes and Slater described graphs \(G\) with \(\gamma_{p}(G)=n\) and also graphs with \(\gamma_{p}(G)=n-1\). In this paper we show all graphs for which \(\gamma_{p}(G)=n-2\).