4,461 research outputs found

    On the diameter of dot-critical graphs

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    A graph G is kk-dot-critical (totaly kk-dot-critical) if GG is dot-critical (totaly dot-critical) and the domination number is kk. In the paper [T. Burtona, D. P. Sumner, Domination dot-critical graphs, Discrete Math, 306 (2006), 11-18] the following question is posed: What are the best bounds for the diameter of a kk-dot-critical graph and a totally kk-dot-critical graph GG with no critical vertices for k4k \geq 4? We find the best bound for the diameter of a kk-dot-critical graph, where k{4,5,6}k \in\{4,5,6\} and we give a family of kk-dot-critical graphs (with no critical vertices) with sharp diameter 2k32k-3 for even k4k \geq 4

    Total Domination Dot Critical and Dot Stable Graphs.

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    Two vertices are said to be identifed if they are combined to form one vertex whose neighborhood is the union of their neighborhoods. A graph is total domination dot-critical if identifying any pair of adjacent vertices decreases the total domination number. On the other hand, a graph is total domination dot-stable if identifying any pair of adjacent vertices leaves the total domination number unchanged. Identifying any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most two. Among other results, we characterize total domination dot-critical trees with total domination number three and all total domination dot-stable graphs

    Protecting a Graph with Mobile Guards

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    Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

    GENERATING DSS GRAPH BY EDGE SUBDIVISION AND EDGE CONTRACTION

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    A graph G is said to be domination subdivision stable ( DSS ) if g ( Gsd uv ) = g ( G ), for all u, v ÃŽ V ( G ), u adjacent to v. In this paper we have provided two methods of obtaining a DSS graph from a non DSS graph

    DETERMINATION OF THE RESTRAINED DOMINATION NUMBER ON VERTEX AMALGAMATION AND EDGE AMALGAMATION OF THE PATH GRAPH WITH THE SAME ORDER

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    Graph theory is a mathematics section that studies discrete objects. One of the concepts studied in graph theory is the restrained dominating set which aims to find the restrained dominating number. This research was conducted by examining the graph operation result of the vertex and edges amalgamation of the path graph in the same order. The method used in this research is the deductive method by using existing theorems to produce new theorems that will be proven deductively true. This research aimed to determine the restrained dominating number in vertex and edges amalgamation of the path graph in the same order. The results obtained from this study are in the form of the theorem about the restrained dominating number of vertex and edges amalgamation of the path graph in the same order, namely: for , ⌋, and for , ⌋

    Domination changing and unchanging signed graphs upon the vertex removal

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    A subset S of V (Σ) is a dominating set of Σ if |N⁺(v) ∩ S| > |N⁻(v) ∩ S| for all v ∈ V − S. This article is to start a study of those signed graphs that are stable and critical in the following way: If the removal of an arbitrary vertex does not change the domination number, the signed graph will be stable. The signed graph, on the other hand, is unstable if an arbitrary vertex is removed and the domination number changes. Specifically, we analyze the change in the domination of the vertex deletion and stable signed graphs.Publisher's Versio

    Observational constraints on interacting quintessence models

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    We determine the range of parameter space of Interacting Quintessence Models that best fits the recent WMAP measurements of Cosmic Microwave Background temperature anisotropies. We only consider cosmological models with zero spatial curvature. We show that if the quintessence scalar field decays into cold dark matter at a rate that brings the ratio of matter to dark energy constant at late times,the cosmological parameters required to fit the CMB data are: \Omega_x = 0.43 \pm 0.12, baryon fraction \Omega_b = 0.08 \pm 0.01, slope of the matter power spectrum at large scals n_s = 0.98 \pm 0.02 and Hubble constant H_0 = 56 \pm 4 km/s/Mpc. The data prefers a dark energy component with a dimensionless decay parameter c^2 =0.005 and non-interacting models are consistent with the data only at the 99% confidence level. Using the Bayesian Information Criteria we show that this exra parameter fits the data better than models with no interaction. The quintessence equation of state parameter is less constrained; i.e., the data set an upper limit w_x \leq -0.86 at the same level of significance. When the WMAP anisotropy data are combined with supernovae data, the density parameter of dark energy increases to \Omega_x \simeq 0.68 while c^2 augments to 6.3 \times 10^{-3}. Models with quintessence decaying into dark matter provide a clean explanation for the coincidence problem and are a viable cosmological model, compatible with observations of the CMB, with testable predictions. Accurate measurements of baryon fraction and/or of matter density independent of the CMB data, would support/disprove these models.Comment: 16 pages, Revtex4, 5 eps figures, to appear in Physical Review
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