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### Graphs with most number of three point induced connected subgraphs

F.T. Boesch, X. Li, J. Rodriguez
1995 Discrete Applied Mathematics
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.  ...

### Algorithms for enumerating connected induced subgraphs of a given order [article]

Shanshan Wang, Chenglong Xiao
2021 arXiv   pre-print
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.  ...

### Some perfect coloring properties of graphs

Claude A Christen, Stanley M Selkow
1979 Journal of combinatorial theory. Series B (Print)
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.  ...

### Triangulations without pointed spanning trees

Oswin Aichholzer, Clemens Huemer, Hannes Krasser
2008 Computational geometry
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.  ...

### On uniquely colorable planar graphs

Gary Chartrand, Dennis P. Geller
1969 Journal of Combinatorial Theory
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  ...

### New Algorithms for Edge Induced König-Egerváry Subgraph Based on Gallai-Edmonds Decomposition

Qilong Feng, Guanlan Tan, Senmin Zhu, Bin Fu, Jianxin Wang, Michael Wagner
2018 International Symposium on Algorithms and Computation
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.  ...

### Nonpositive eigenvalues of the adjacency matrix and lower bounds for Laplacian eigenvalues

Zachary B. Charles, Miriam Farber, Charles R. Johnson, Lee Kennedy-Shaffer
2013 Discrete Mathematics
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.  ...

### The very small world of the well-connected

Xiaolin Shi, Matthew Bonner, Lada Adamic, Anna C. Gilbert
2009 ACM SIGWEB Newsletter
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  ...

### The very small world of the well-connected

Xiaolin Shi, Matthew Bonner, Lada A. Adamic, Anna C. Gilbert
2008 Proceedings of the nineteenth ACM conference on Hypertext and hypermedia - HT '08
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  ...

### Parameterized complexity of connected even/odd subgraph problems

Fedor V. Fomin, Petr A. Golovach
2014 Journal of computer and system sciences (Print)
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.  ...

### Graphs with forbidden subgraphs

Gary Chartrand, Dennis Geller, Stephen Hedetniemi
1971 Journal of combinatorial theory. Series B (Print)
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.  ...

### Metric subgraphs of the chamfer metrics and the Melter-Tomescu path generated metrics

Frank Rhodes
1995 Discrete Mathematics
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.  ...

### Nonpositive Eigenvalues of the Adjacency Matrix and Lower Bounds for Laplacian Eigenvalues [article]

Zachary B. Charles, Miriam Farber, Charles R. Johnson, Lee Kennedy-Shaffer
2012 arXiv   pre-print
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.  ...

### On the Rank Number of Grid Graphs [article]

Sitan Chen
2013 arXiv   pre-print
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.  ...

### SkyGraph: an algorithm for important subgraph discovery in relational graphs

Apostolos N. Papadopoulos, Apostolos Lyritsis, Yannis Manolopoulos
2008 Data mining and knowledge discovery
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  ...
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