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### Making the components of a graph k-connected

V. Nikiforov, R.H. Schelp
2007 Discrete Applied Mathematics
For every integer k 2 and graph G, consider the following natural procedure: if G has a component G that is not k-connected, remove G if |G | k, otherwise remove a cutset U ⊂ V (G ) with |U | < k; do the  ...  same with the remaining graph until only k-connected components are left or all vertices are removed.  ...  Given an integer k 2 and a graph G, a k-cutting procedure of G is defined as follows: while G has a component G that is not k-connected and V (G) = ∅ do if G has a component G with |G | k then set G =  ...

### Making Bidirected Graphs Strongly Connected [article]

Tatsuya Matsuoka, Shun Sato
2017 arXiv   pre-print
We consider problems to make a given bidirected graph strongly connected with minimum cardinality of additional signs or additional arcs.  ...  For the former problem, we show the minimum number of additional signs and give a linear-time algorithm for finding an optimal solution.  ...  Related Works It is obvious that the minimum number of additional edges to make a given undirected graph connected is fewer than the number of connected components of a given graph by one.  ...

### Conflict-free connection numbers of line graphs [article]

Bo Deng, Wenjing Li, Xueliang Li, Yaping Mao, Haixing Zhao
2017 arXiv   pre-print
For a connected graph G, the conflict-free connection number of G, denoted by cfc(G), is defined as the minimum number of colors that are required to make G conflict-free connected.  ...  Secondly, we get the exact value of the conflict-free connection number of a connected claw-free graph, especially a connected line graph.  ...  For a connected graph G, the rainbow connection number of G, denoted by rc(G), is defined as the minimum number of colors that are needed to make G rainbow connected.  ...

### On the maximum value of conflict-free verex-connection number of graphs [article]

Zhenzhen Li, Baoyindureng Wu
2017 arXiv   pre-print
A vertex-colored graph is said to be conflict-free vertex-connected if any two vertices of the graph are connected by a conflict-free path.  ...  The conflict-free vertex-connection number, denoted by vcfc(G), is defined as the smallest number of colors required to make G conflict-free vertex-connected.  ...  The conflict-free connection number of a connected graph, denoted by cf c(G), is defined a the smallest number of colors required to make G conflict-free connected.  ...

### Connectivity structure of bipartite graphs via the KNC-plot

Ravi Kumar, Andrew Tomkins, Erik Vee
2008 Proceedings of the international conference on Web search and web data mining - WSDM '08
The KNC-plot shows the degradation of connectivity of the graph as a function of k.  ...  In this paper we introduce the k-neighbor connectivity plot, or KNC-plot, as a tool to study the macroscopic connectivity structure of sparse bipartite graphs.  ...  We extract the size of the largest component and the number of connected components in the graph G k , as a function of k, and plots the results in the KNC-plot.  ...

### Minimum 2-vertex-twinless connected spanning subgraph problem [article]

Raed Jaberi
2020 arXiv   pre-print
Given a 2-vertex-twinless connected directed graph G=(V,E), the minimum 2-vertex-twinless connected spanning subgraph problem is to find a minimum cardinality edge subset E^t⊆ E such that the subgraph  ...  Let G^1 be a minimal 2-vertex-connected subgraph of G.  ...  The results of [2, 10] imply that every minimal 2-vertex-connected spanning subgraph has at most 4n edges.  ...

### Local Reconstructors and Tolerant Testers for Connectivity and Diameter [chapter]

Andrea Campagna, Alan Guo, Ronitt Rubinfeld
2013 Lecture Notes in Computer Science
For this model, we achieve local property reconstructors for the properties of connectivity and k-connectivity in undirected graphs, and the property of strong connectivity in directed graphs.  ...  matrix of a "correction" of the graph, i.e. a graph which has the property and is close to the given graph.  ...  Ideally, we have exactly one special vertex in each connected component, since this number of edges is both necessary and sufficient to make the graph connected.  ...

### Local reconstructors and tolerant testers for connectivity and diameter [article]

Andrea Campagna, Alan Guo, Ronitt Rubinfeld
2013 arXiv   pre-print
For this model, we achieve local property reconstructors for the properties of connectivity and k-connectivity in undirected graphs, and the property of strong connectivity in directed graphs.  ...  matrix of a "correction" of the graph, i.e. a graph which has the property and is close to the given graph.  ...  Ideally, we have exactly one special vertex in each connected component, since this number of edges is both necessary and sufficient to make the graph connected.  ...

### One-Way Trail Orientations [article]

Anders Aamand, Niklas Hjuler, Jacob Holm, Eva Rotenberg
2017 arXiv   pre-print
Monthly, 1980] for mixed multigraphs states that the undirected edges of a mixed multigraph can be oriented making the resulting directed graph strongly connected exactly when the mixed graph is connected  ...  Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected? Robbins' theorem [Robbins, Am. Math.  ...  Let a 1 , . . . , a k be the number of vertices in the "large" 3-edge connected components.  ...

### Page 6543 of Mathematical Reviews Vol. , Issue 93m [page]

1993 Mathematical Reviews
This makes the operation applica- ble to hypergraphs and makes possible a recursive definition of k-connected components (which obviously are not the traditional ones in relation to the original graph)  ...  The author does not make contact with Matula’s concepts of k- components and k-blocks and good algorithms for their detection, which are nearer to the traditional theory [see, e.g., D. W.  ...

### Tight query complexity bounds for learning graph partitions [article]

Xizhi Liu, Sayan Mukherjee
2022 arXiv   pre-print
Given a partition of a graph into connected components, the membership oracle asserts whether any two vertices of the graph lie in the same component or not.  ...  We prove that for n≥ k≥ 2, learning the components of an n-vertex hidden graph with k components requires at least (k-1)n- k2 membership queries.  ...  Research of the first author was supported by ERC Advanced Grant 101020255.  ...

### Chromatic classes of 2-connected (n,n+4)-graphs with at least four triangles

Y.H Peng, G.C Lau
2004 Discrete Mathematics
Notation and basic results The following are some useful known results and techniques for determining the chromatic polynomial of a graph.  ...  In this paper, we shall determine all the chromatic equivalence classes of 2-connected (n, n + 4)graphs with at least four triangles.  ...  (iv) The graph J 18 is a disconnected (n, n + 2)-graph having four components. If three of the components are connected by two paths, we get a disconnected (n, n + 4)-graph.  ...

### The Topology of Competitively Constructed Graphs

Alan Frieze, Wesley Pegden
2014 Electronic Journal of Combinatorics
We show a sharp topological threshold for this game: for the case \$k=3\$ a player can ensure the resulting graph is planar, while for the case \$k=4\$, a player can force the appearance of arbitrarily large  ...  We consider a simple game, the \$k\$-regular graph game, in which players take turns adding edges to an initially empty graph subject to the constraint that the degrees of vertices cannot exceed \$k\$.  ...  Thus there is no surface S for which a player of the 4-regular graph game can ensure that the connected components of the result has a drawing on S.  ...

### New Clusterization Method Based on Graph Connectivity Search

Michael G. Sadovsky, Eugene Yu. Bushmelev, Anatoly N. Ostylovsky
2017 Journal of Siberian Federal University Mathematics & Physics
The method is based on a sequential elimination of the longest distances in dataset, so that the relevant graph looses some edges. The method stops when the graph becomes disconnected.  ...  Hence, an examination for graph connectivity must be changed for the connected components search, in the graph. There is a number of algorithms to seek for connected components, in a graph.  ...  Indeed, any graph is unambiguously decomposed into a number of connected components; a connected graph has the unique connected component, and vice versa    .  ...

### Conflict-free connection number and independence number of a graph [article]

Jing Wang, Meng Ji
2020 arXiv   pre-print
The conflict-free connection number of a connected graph G, denoted by cfc(G), is defined as the minimum number of colors that are required in order to make G conflict-free connected.  ...  In this paper, we investigate the relation between the conflict-free connection number and the independence number of a graph.  ...  Acknowledgement 1 This work was done during the first author was visiting Nankai University.  ...
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