• Title/Summary/Keyword: k-connected graph

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Connected geodesic number of a fuzzy graph

  • Rehmani, Sameeha;Sunitha, M.S.
    • Annals of Fuzzy Mathematics and Informatics
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    • v.16 no.3
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    • pp.301-316
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    • 2018
  • In this paper, the concept of connected geodesic number, $gn_c(G)$, of a fuzzy graph G is introduced and its limiting bounds are identified. It is proved that all extreme nodes of G and all cut-nodes of the underlying crisp graph $G^*$ belong to every connected geodesic cover of G. The connected geodesic number of complete fuzzy graphs, fuzzy cycles, fuzzy trees and of complete bipartite fuzzy graphs are obtained. It is proved that for any pair k, n of integers with $3{\leq}k{\leq}n$, there exists a connected fuzzy graph G : (V, ${\sigma}$, ${\mu}$) on n nodes such that $gn_c(G)=k$. Also, for any positive integers $2{\leq}a<b{\leq}c$, it is proved that there exists a connected fuzzy graph G : (V, ${\sigma}$, ${\mu}$) such that the geodesic number gn(G) = a and the connected geodesic number $gn_c(G)=b$.

EXISTENCE OF SPANNING 3-TREES IN A 3-CONNECTED LOCALLY FINITE VAP-FREE PLANE GRAPH

  • Jung, Hwan-Ok
    • Journal of applied mathematics & informatics
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    • v.28 no.3_4
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    • pp.893-908
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    • 2010
  • In this paper we prove the existence of spanning 3-trees in a 3-connected infinite locally finite VAP-free plane graph. Together with the results of Barnette and the author, this yields that every finite or infinite 3-connected locally finite VAP-free plane graph contains a spanning 3-tree.

THE CONNECTED DOUBLE GEODETIC NUMBER OF A GRAPH

  • SANTHAKUMARAN, A.P.;JEBARAJ, T.
    • Journal of applied mathematics & informatics
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    • v.39 no.1_2
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    • pp.155-163
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    • 2021
  • For a connected graph G of order n, a set S of vertices is called a double geodetic set of G if for each pair of vertices x, y in G there exist vertices u, v ∈ S such that x, y ∈ I[u, v]. The double geodetic number dg(G) is the minimum cardinality of a double geodetic set. Any double godetic set of cardinality dg(G) is called a dg-set of G. A connected double geodetic set of G is a double geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected double geodetic set of G is the connected double geodetic number of G and is denoted by dgc(G). A connected double geodetic set of cardinality dgc(G) is called a dgc-set of G. Connected graphs of order n with connected double geodetic number 2 or n are characterized. For integers n, a and b with 2 ≤ a < b ≤ n, there exists a connected graph G of order n such that dg(G) = a and dgc(G) = b. It is shown that for positive integers r, d and k ≥ 5 with r < d ≤ 2r and k - d - 3 ≥ 0, there exists a connected graph G of radius r, diameter d and connected double geodetic number k.

ON THE MINIMUM ORDER OF 4-LAZY COPS-WIN GRAPHS

  • Sim, Kai An;Tan, Ta Sheng;Wong, Kok Bin
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1667-1690
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    • 2018
  • We consider the minimum order of a graph G with a given lazy cop number $c_L(G)$. Sullivan, Townsend and Werzanski [7] showed that the minimum order of a connected graph with lazy cop number 3 is 9 and $k_3{\square}k_3$ is the unique graph on nine vertices which requires three lazy cops. They conjectured that for a graph G on n vertices with ${\Delta}(G){\geq}n-k^2$, $c_L(G){\leq}k$. We proved that the conjecture is true for k = 4. Furthermore, we showed that the Petersen graph is the unique connected graph G on 10 vertices with ${\Delta}(G){\leq}3$ having lazy cop number 3 and the minimum order of a connected graph with lazy cop number 4 is 16.

Conditions for Disjoint Path Coverability in Proper Interval Graphs (진구간 그래프의 서로소인 경로 커버에 대한 조건)

  • Park, Jung-Heum
    • Journal of KIISE:Computer Systems and Theory
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    • v.34 no.10
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    • pp.539-554
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    • 2007
  • In this Paper, we investigate conditions for proper interval graphs to have k-disjoint path covers of three types each: one-to-one, one-to-many, and many-to-many. It was proved that for $k{\geq}2$, a proper interval graph is one-to-one k-disjoint path coverable if and only if the graph is k-connected, and is one-to-many k-disjoint path coverable if and only if the graph is k+1-connected. For $k{\geq}3$, a Proper interval graph is (paired) many-to-many k-disjoint path coverable if and only if the graph is 2k-1-connected.

A New Connected Coherence Tree Algorithm For Image Segmentation

  • Zhou, Jingbo;Gao, Shangbing;Jin, Zhong
    • KSII Transactions on Internet and Information Systems (TIIS)
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    • v.6 no.4
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    • pp.1188-1202
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    • 2012
  • In this paper, we propose a new multi-scale connected coherence tree algorithm (MCCTA) by improving the connected coherence tree algorithm (CCTA). In contrast to many multi-scale image processing algorithms, MCCTA works on multiple scales space of an image and can adaptively change the parameters to capture the coarse and fine level details. Furthermore, we design a Multi-scale Connected Coherence Tree algorithm plus Spectral graph partitioning (MCCTSGP) by combining MCCTA and Spectral graph partitioning in to a new framework. Specifically, the graph nodes are the regions produced by CCTA and the image pixels, and the weights are the affinities between nodes. Then we run a spectral graph partitioning algorithm to partition on the graph which can consider the information both from pixels and regions to improve the quality of segments for providing image segmentation. The experimental results on Berkeley image database demonstrate the accuracy of our algorithm as compared to existing popular methods.

THE OUTER-CONNECTED VERTEX EDGE DOMINATION NUMBER OF A TREE

  • Krishnakumari, Balakrishna;Venkatakrishnan, Yanamandram Balasubramanian
    • Communications of the Korean Mathematical Society
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    • v.33 no.1
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    • pp.361-369
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    • 2018
  • For a given graph G = (V, E), a set $D{\subseteq}V(G)$ is said to be an outer-connected vertex edge dominating set if D is a vertex edge dominating set and the graph $G{\backslash}D$ is connected. The outer-connected vertex edge domination number of a graph G, denoted by ${\gamma}^{oc}_{ve}(G)$, is the cardinality of a minimum outer connected vertex edge dominating set of G. We characterize trees T of order n with l leaves, s support vertices, for which ${\gamma}^{oc}_{ve}(T)=(n-l+s+1)/3$ and also characterize trees with equal domination number and outer-connected vertex edge domination number.

SUFFICIENT CONDITIONS FOR SOME HAMILTONIAN PROPERTIES AND K-CONNECTIVITY OF GRAPHS

  • LI, RAO
    • Journal of applied mathematics & informatics
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    • v.34 no.3_4
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    • pp.221-225
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    • 2016
  • For a connected graph G = (V, E), its inverse degree is defined as $\sum_{{\upsilon}{\in}{V}}^{}\frac{1}{d(\upsilon)}$. Using an upper bound for the inverse degree of a graph obtained by Cioabă in [4], we in this note present sufficient conditions for some Hamiltonian properties and k-connectivity of a graph.

Cross-index of a Graph

  • Kawauchi, Akio;Shimizu, Ayaka;Yaguchi, Yoshiro
    • Kyungpook Mathematical Journal
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    • v.59 no.4
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    • pp.797-820
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    • 2019
  • For every tree T, we introduce a topological invariant, called the T-cross-index, for connected graphs. The T-cross-index of a graph is a non-negative integer or infinity according to whether T is a tree basis of the graph or not. It is shown how this cross-index is independent of the other topological invariants of connected graphs, such as the Euler characteristic, the crossing number and the genus.

GENERALIZED CAYLEY GRAPHS OF RECTANGULAR GROUPS

  • ZHU, YONGWEN
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.4
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    • pp.1169-1183
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    • 2015
  • We describe generalized Cayley graphs of rectangular groups, so that we obtain (1) an equivalent condition for two Cayley graphs of a rectangular group to be isomorphic to each other, (2) a necessary and sufficient condition for a generalized Cayley graph of a rectangular group to be (strong) connected, (3) a necessary and sufficient condition for the colour-preserving automorphism group of such a graph to be vertex-transitive, and (4) a sufficient condition for the automorphism group of such a graph to be vertex-transitive.