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Graphs with most number of three point induced connected subgraphs
1995
Discrete Applied Mathematics
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Let G be a simple graph with e perfectly reliable edges and n nodes which fail independently and with the same probability p. The residual connectedness reliability R ...
Let tj(G), where 0 <j < 3, be the number of 3-subsets of [n] that induce a subgraph in G having exactly j edges. Let S,(G) be the number of 3-subsets of [n] that induce a connected subgraph in G. ...
The residual connectedness reliability of a network G, denoted by R(G,p), is the probability that the graph induced by the surviving points is connected. ...
doi:10.1016/0166-218x(93)e0155-r
fatcat:7gmfhyud4bhclhozh7xrlbb23e
Algorithms for enumerating connected induced subgraphs of a given order
[article]
2021
arXiv
pre-print
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2021 pre-print arXiv:2112.07197v1 fatcat:qn563sf5tbcwdpqk75dk5qdti4
In this paper, we propose two algorithms for enumerating all connected induced subgraphs of a given order from connected undirected graphs. ...
Enumerating all connected subgraphs of a given order from graphs is a computationally challenging task. ...
Similar to [7], we group the
benchmark graphs into three subsets: small graphs with 𝑛 < 500, medium-size subgraphs with 500 ≤ 𝑛 < 5000
and large graphs with 𝑛 ≥ 5000. ...
arXiv:2112.07197v2
fatcat:d5ijgzx3n5gdjapwhje326t42i
Some perfect coloring properties of graphs
1979
Journal of combinatorial theory. Series B (Print)
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Paths with at least four points are connected examples with y = 3 and $ arbitrarily large (the only connected graphs with y = 2 are the complete bipartite graphs, for which # = 2). ...
A graph G is yx-perfect zf and only if for each induced subgraph H of G, y(H) + y(a 6 II H 11 + 1, where 11 H II denotes the number of points of H. Proof. ...
doi:10.1016/0095-8956(79)90067-4
fatcat:hvrz35tav5gxjj5giack7lud5a
Triangulations without pointed spanning trees
2008
Computational geometry
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Problem 50 in the Open Problems Project of the computational geometry community asks whether any triangulation on a point set in the plane contains a pointed spanning tree as a subgraph. ...
As a consequence we show that there exist triangulations which require a linear number of edge flips to become Hamiltonian. ...
Acknowledgements The innocent looking conjecture that any triangulation contains a pointed spanning tree as a subgraph has fascinated several people. ...
doi:10.1016/j.comgeo.2007.07.006
fatcat:qomntjn5v5ez7lx4x4kz7zgc2e
On uniquely colorable planar graphs
1969
Journal of Combinatorial Theory
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A labeled graph G with chromatic number n is called uniquely n-colorable or simply uniquely colorable if every two partitions of the point set of G into n color classes are the same. ...
In particular, it is shown that uniquely 3-colorable planar graphs with at least four points contain at least two triangles, uniquely 4-colorable planar graphs are maximal planar, and uniquely 5-colorable ...
ACKNOWLEDGMENT The authors were introduced to the concept of unique colorability by Dorwin Cartwright of the Research Center for Group Dynamics at the University of Michigan, and it is to Professor Cartwright ...
doi:10.1016/s0021-9800(69)80087-6
fatcat:sum4isfuy5hodksgswatq3akca
New Algorithms for Edge Induced König-Egerváry Subgraph Based on Gallai-Edmonds Decomposition
2018
International Symposium on Algorithms and Computation
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We obtain a new structural property of König-Egerváry subgraph: every graph G = (V, E) has an edge induced König-Egerváry subgraph with at least 2|E|/3 edges. ...
Using 2|E|/3 as a lower bound, we define the Edge Induced König Subgraph above lower bound problem, and give a kernel of at most 30k edges for the problem. ...
Maximum Edge Induced König Subgraph: Given a graph G = (V, E), find a set E ⊆ E with maximum number of edges such that the edges in E induce a König subgraph. ...
doi:10.4230/lipics.isaac.2018.31
dblp:conf/isaac/FengTZFW18
fatcat:iuwlebsdzjgxzmibtuvflpzuy4
Nonpositive eigenvalues of the adjacency matrix and lower bounds for Laplacian eigenvalues
2013
Discrete Mathematics
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Let NPO(k) be the smallest number n such that the adjacency matrix of any undirected graph with n vertices or more has at least k nonpositive eigenvalues. ...
In addition, we prove that for all k ≥ 5, R(k, k + 1) ≥ NPO(k) > T k , in which R(k, k + 1) is the Ramsey number for k and k + 1, and T k is the kth triangular number. ...
The Ramsey number R(m, n) is the minimum number of vertices such that all graphs of order R(m, n) or more have either an independent set of size m or a complete graph of order n as an induced subgraph. ...
doi:10.1016/j.disc.2013.03.010
fatcat:fdbwhfdvr5ds5h2ycmez7xwypy
The very small world of the well-connected
2009
ACM SIGWEB Newsletter
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We show that graph synopses defined by the importance of vertices provide small, relatively accurate portraits, independent of the importance measure, of the larger underlying graphs and of the important ...
Frequently, small sets of vertices dominate various graph and statistical properties of these networks and, because of this, they are relevant for structural analysis and efficient algorithms and engineering ...
Figure 9 : 9 The number of edges between important vertices, where importance is measured by degree, in three networks: 1) power law network with α = 2.2, n = 1000, 2) Erdös-Renyi graph with the same average ...
doi:10.1145/1457507.1457511
fatcat:gl5drf2c5bghzibamw6jqgjpnm
The very small world of the well-connected
2008
Proceedings of the nineteenth ACM conference on Hypertext and hypermedia - HT '08
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We show that graph synopses defined by the importance of vertices provide small, relatively accurate portraits, independent of the importance measure, of the larger underlying graphs and of the important ...
Frequently, small sets of vertices dominate various graph and statistical properties of these networks and, because of this, they are relevant for structural analysis and efficient algorithms and engineering ...
Figure 9 : 9 The number of edges between important vertices, where importance is measured by degree, in three networks: 1) power law network with α = 2.2, n = 1000, 2) Erdös-Renyi graph with the same average ...
doi:10.1145/1379092.1379108
dblp:conf/ht/ShiBAG08
fatcat:yfneoiww6bcfjmnn3rwmyiamqy
Parameterized complexity of connected even/odd subgraph problems
2014
Journal of computer and system sciences (Print)
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Subgraph; and a connected k-vertex induced subgraph with all vertices of even degrees, the problem known as k-Vertex Eulerian Subgraph. ...
For a given graph G and integer k, the task is to decide if G contains a (connected) subgraph with k vertices (edges) with all vertices of even (odd) degrees. ...
This means that when G has a connected odd subgraph H with k edges containing r, then there is a connected odd subgraph H with k edges containing r and of treewidth at most three. ...
doi:10.1016/j.jcss.2013.07.002
fatcat:mdxidlkmdvhmvcvneg7rbg2i7y
Graphs with forbidden subgraphs
1971
Journal of combinatorial theory. Series B (Print)
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For example, a tree can be defined as a connected graph which contains no cycles, and Kuratowski [22] characterized planar graphs as those graphs which fail to contain subgraphs homeomorphic from the complete ...
Many graphs which are encountered in the study of graph theory are characterized by a type of configuration or subgraph they possess. ...
If all three points are colored the same, say cy, then, by coloring u with /3, the subgraph of G induced by the points colored /3 is clearly a forest. ...
doi:10.1016/0095-8956(71)90065-7
fatcat:twjplw5ainfv7mc4qshnieyepu
Metric subgraphs of the chamfer metrics and the Melter-Tomescu path generated metrics
1995
Discrete Mathematics
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The results for chamfer metrics are related to corresponding results for the metrics generated by the two-, three-and four-direction graphs studied by Melter and Tomescu. ...
rays of the cone. ...
The contradiction proves the sufficiency of the conditions for the four-connection graph. Now let o~f' be a connected induced subgraph of (7/2, T84) which is c/a-convex and d/b-convex. ...
doi:10.1016/0012-365x(94)00238-e
fatcat:a3z43kcvyjd35ingrftjaqxb54
Nonpositive Eigenvalues of the Adjacency Matrix and Lower Bounds for Laplacian Eigenvalues
[article]
2012
arXiv
pre-print
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Let NPO(k) be the smallest number n such that the adjacency matrix of any undirected graph with n vertices or more has at least k nonpositive eigenvalues. ...
In addition, we prove that for all k ≥ 5, R(k,k+1) > NPO(k) > T_k, in which R(k,k+1) is the Ramsey number for k and k+1, and T_k is the k^th triangular number. ...
The Ramsey number R(m, n) is the minimum number of vertices such that all graphs of order R(m, n) or more have either an independent set of size m or a complete graph of order n as an induced subgraph. ...
arXiv:1108.4810v3
fatcat:7hp2jw7lqzhozdyezbl5mrtnuq
On the Rank Number of Grid Graphs
[article]
2013
arXiv
pre-print
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A vertex k-ranking is a labeling of the vertices of a graph with integers from 1 to k so any path connecting two vertices with the same label will pass through a vertex with a greater label. ...
The rank number of a graph is defined to be the minimum possible k for which a k-ranking exists for that graph. For mxn grid graphs, the rank number has been found only for m<4. ...
If we remove the other three points in two columns, we induce a G 4, n−2 2 subgraph, and by removing the remaining point in the cut set, we remove its corner. ...
arXiv:1208.1814v3
fatcat:vcuq7woqnnfvbdozrrvxr3uul4
SkyGraph: an algorithm for important subgraph discovery in relational graphs
2008
Data mining and knowledge discovery
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The importance of a subgraph is determined by: (i) the order of the subgraph (the number of vertices) and (ii) the subgraph edge Responsible editors: 123 58 A. N. Papadopoulos et al. connectivity. ...
However, in some cases the number of mined patterns is large, posing difficulties in selecting the most important ones. ...
are grateful to Kostas Tsichlas and Anastasios Gounaris for their helpful comments, Xifeng Yan for providing the microarray data and the anonymous reviewers for their aid towards improving the quality of ...
doi:10.1007/s10618-008-0109-y
fatcat:ijowotahqrdlrlawtw3idy744u
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