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1.
A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The minimum cardinality of a paired-dominating set of G is the paired-domination number of G, denoted by pr(G). If G does not contain a graph F as an induced subgraph, then G is said to be F-free. In particular if F=K1,3 or K4–e, then we say that G is claw-free or diamond-free, respectively. Let G be a connected cubic graph of order n. We show that (i) if G is (K1,3,K4–e,C4)-free, then pr(G)3n/8; (ii) if G is claw-free and diamond-free, then pr(G)2n/5; (iii) if G is claw-free, then pr(G)n/2. In all three cases, the extremal graphs are characterized.Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal. This paper was written while the second author was visiting the Laboratoire de Recherche en Informatique (LRI) at the Université de Paris-Sud in July 2002. The second author thanks the LRI for their warm hospitality 相似文献
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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, denoted by γ pr (G), is the minimum cardinality of a paired-dominating set in G. In this paper we investigate the paired-domination number in claw-free graphs. Specifically, we show that γ pr (G) ≤ (3n ? 1)/5 if G is a connected claw-free graph of order n with minimum degree at least three and that this bound is sharp. 相似文献
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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 γ
pr
(G), is the minimum cardinality of a paired-dominating set of G. The paired-domination subdivision number
sd
γpr
(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. In this paper we establish upper bounds
on the paired-domination subdivision number and pose some problems and conjectures. 相似文献
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A set \(S\subseteq V\) is a paired-dominating set if every vertex in \(V{\setminus } S\) 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 \(\gamma _{pr}(G)\), 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 \(\delta (G)\ge 3\), then \(\gamma _{pr}(G)\le 4n/7\). In this paper, we confirm this conjecture for k-regular graphs with \(k\ge 4\). 相似文献
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A paired-dominating set of a graph 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. Recently, Chen et al. (Acta Math
Sci Ser A Chin Ed 27(1):166–170, 2007) proved that a cubic graph has paired-domination number at most three-fifths the number
of vertices in the graph. In this paper, we show that the Petersen graph is the only connected cubic graph with paired-domination
number three-fifths its order. 相似文献
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A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by , is the minimum cardinality of a paired-dominating set of G. In [1], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, the critical cases
are not claws but subdivided stars. We here give a bound for graphs containing no induced P
5, which seems to be the critical case. 相似文献
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Noga Alon 《Journal of Graph Theory》1994,18(4):343-347
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A.V. Pyatkin 《Discrete Mathematics》2008,308(9):1749-1750
A graph G=(V,E) is an integral sum graph (ISG) if there exists a labeling S(G)⊂Z such that V=S(G) and for every pair of distinct vertices u,v∈V, uv is an edge if and only if u+v∈V. A vertex in a graph is called a fork if its degree is not 2. In 1998, Chen proved that every tree whose forks are at distance at least 4 from each other is an ISG. In 2004, He et al. reduced the distance to 3. In this paper we reduce the distance further to 2, i.e. we prove that every tree whose forks are at least distance 2 apart is an ISG. 相似文献
9.
Ruth Haas 《Acta Appl Math》1998,51(2):113-122
Let Sr() be the module of all splines of smoothness r on the rectilinear partition which subdivides some domain D. Further, let Sr() be the module of all splines of smoothness r on which also subdivides D, where is a finer subdivision of . We study the relationship between a generating set of Sr() and a generating set for Sr(). This paper gives an algorithm for extending a generating set for Sr() to one for for Sr(). This method is built on algebraic properties of splines and the Gröbner Basis Algorithm. 相似文献
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Sanming Zhou 《Monatshefte für Mathematik》2003,139(1):69-81
With any G-symmetric graph Γ admitting a nontrivial G-invariant partition , we may associate a natural “cross-sectional” geometry, namely the 1-design in which for and if and only if α is adjacent to at least one vertex in C, where and is the neighbourhood of B in the quotient graph of Γ with respect to . In a vast number of cases, the dual 1-design of contains no repeated blocks, that is, distinct vertices of B are incident in with distinct subsets of blocks of . The purpose of this paper is to give a general construction of such graphs, and then prove that it produces all of them.
In particular, we show that such graphs can be reconstructed from and the induced action of G on . The construction reveals a close connection between such graphs and certain G-point-transitive and G-block-transitive 1-designs. By using this construction we give a characterization of G-symmetric graphs such that there is at most one edge between any two blocks of . This leads to, in a subsequent paper, a construction of G-symmetric graphs such that and each is incident in with vertices of B.
The work was supported by a discovery-project grant from the Australian Research Council.
Received April 24, 2001; in revised form October 9, 2002
Published online May 9, 2003 相似文献
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Ola Hössjer 《Methodology and Computing in Applied Probability》2014,16(4):777-810
For populations with geographic substructure and selectively neutral genetic data, the short term dynamics is a balance between migration and genetic drift. Before fixation of any allele, the system enters into a quasi equilibrium (QE) state. Hössjer and Ryman (2012) developed a general QE methodology for computing approximations of spatial autocorrelations of allele frequencies between subpopulations, subpopulation differentiation (fixation indexes) and variance effective population sizes. In this paper we treat a class of models with translationally invariant migration and use Fourier transforms for computing these quantities. We show how the QE approach is related to other methods based on conditional kinship coefficients between subpopulations under mutation-migration-drift equilibrium. We also verify that QE autocorrelations of allele frequencies are closely related to the expected value of Moran’s autocorrelation function and treat limits of continuous spatial location (isolation by distance) and an infinite lattice of subpopulations. The theory is illustrated with several examples including island models, circular and torus stepping stone models, von Mises models, hierarchical island models and Gaussian models. It is well known that the fixation index contains information about the effective number of migrants. The spatial autocorrelations are complementary and typically reveal the type of migration (local or global). 相似文献
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王继顺 《数学的实践与认识》2017,(7):152-160
通过揭示完全蛛网图和渔网图的结构特点,研究了它们的邻点可区别I-全染色问题,并运用构造法给出了其邻点可区别I-全染色,从而获得了它们的邻点可区别I-全色数. 相似文献