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The complexity of graph connectivity [chapter]

Avi Wigderson
1992 Lecture Notes in Computer Science  
There seem to be two complimentary reasons for this wide interest in the complexity of graph connectivity.  ...  In this paper we survey the major developments in understanding the complexity of the graph connectivity problem in several computational models, and highlight some challenging open problems.  ...  Acknowledgments I wish to thank Moni Naor, Ilan Newman and Noam Nisan for reading and improving an earlier version of this paper.  ... 
doi:10.1007/3-540-55808-x_10 fatcat:5cjmn5ndo5eg7jyfcyump25mry

The Parameterised Complexity of Counting Connected Subgraphs and Graph Motifs [article]

Mark Jerrum, Kitty Meeks
2014 arXiv   pre-print
We show that exactly counting the number of connected induced k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand there exists an FPTRAS for the problem; more generally, we show  ...  We then apply these results to a counting version of the Graph Motif problem.  ...  ÓÒÒ Ø ÁÒ Ù ËÙ Ö Ô The number of multicoloured connected induced subgraphs in a graph G can be computed by inclusionexclusion from the numbers of connected induced subgraphs in the 2 k subgraphs of G induced  ... 
arXiv:1308.1575v3 fatcat:fryb77k3gze65clacn6kgap4pq

The complexity of optimal design of temporally connected graphs [article]

Eleni C. Akrida, Leszek Gasieniec, George B. Mertzios, Paul G. Spirakis
2016 arXiv   pre-print
We study the design of small cost temporally connected graphs, under various constraints.  ...  We then consider the case in which a designer of temporal graphs can freely choose availability instances for all edges and aims for temporal connectivity with very small cost; the cost is the total number  ...  We resolve here the complexity of finding the maximum number of redundant labels in any given temporal graph.  ... 
arXiv:1502.04579v3 fatcat:jw5d3xrqwbbmhi337gft5ye2wm

The Exponential Time Complexity of Computing the Probability That a Graph Is Connected [chapter]

Thore Husfeldt, Nina Taslaman
2010 Lecture Notes in Computer Science  
We show that for every probability p with 0 < p < 1, computation of all-terminal graph reliability with edge failure probability p requires time exponential in Omega(m/ log^2 m) for simple graphs of m  ...  edges under the Exponential Time Hypothesis.  ...  Since n ≤ m for connected graphs, the result implies the lower bound exp(Ω(n/ log 2 n)) in terms of the parameter n, the number of vertices of the input graph.  ... 
doi:10.1007/978-3-642-17493-3_19 fatcat:mkayelond5ekfdgsf3kz6rdxze

On the complexity of computing the k-restricted edge-connectivity of a graph [article]

Luis Pedro Montejano, Ignasi Sau
2016 arXiv   pre-print
Very recently, in the parameterized complexity community the notion of good edge separation of a graph has been defined, which happens to be essentially the same as the k-restricted edge-connectivity.  ...  The k-restricted edge-connectivity of a graph G, denoted by λ_k(G), is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least k vertices  ...  We would like to thank the anonymous referees for helpful remarks that improved the presentation of the manuscript.  ... 
arXiv:1502.07659v2 fatcat:jbak456vqbhrno54sb7ypns7am

On Connectivity of the Facet Graphs of Simplicial Complexes [article]

Ilan I. Newman, Yuri Rabinovich
2015 arXiv   pre-print
The paper studies the connectivity properties of facet graphs of simplicial complexes of combinatorial interest.  ...  It is also shown that the facet graph of a d-cycle cannot be split into more than s connected components by removing at most s vertices.  ...  Theorem 4.1 has an immediate implication on connectivity of the facet graphs of d-biconnected sets S of d-complexes.  ... 
arXiv:1502.02232v1 fatcat:mpvzuxqatfccfhbt25aju77ke4

On the complexity of structure and substructure connectivity of graphs [article]

Huazhong Lü, Tingzeng Wu
2021 arXiv   pre-print
In this paper, we characterize the complexity of determining structure and substructure connectivity of graphs, showing that they are both NP-complete.  ...  The connectivity of a graph is an important parameter to measure its reliability. Structure and substructure connectivity are two novel generalizations of the connectivity.  ...  This completes the proof. Remark. The authors [2] constructed the similar graph G ′ as in Theorem 2 to prove NP-completeness of neighbor connectivity by reducing from dominating set problem.  ... 
arXiv:2110.05917v1 fatcat:rsnsdksojfgf7kli55n7mewube

On the Eccentric Connectivity Polynomial of ℱ-Sum of Connected Graphs

Muhammad Imran, Shehnaz Akhter, Zahid Iqbal
2020 Complexity  
The eccentric connectivity polynomial (ECP) of a connected graph G=VG,EG is described as ξcG,y=∑a∈VGdegGayecGa, where ecGa and degGa represent the eccentricity and the degree of the vertex a, respectively  ...  In this article, we work out the ECP of ℱ-sum of graphs.  ...  Acknowledgments is research was supported by UPAR Grant of United Arab Emirates University (UAEU), Al Ain, UAE, via Grant nos. G00002590 and G00003271.  ... 
doi:10.1155/2020/5061682 fatcat:cammxnm6tzhl7lc73lgukyuvy4

The complexity of determining the rainbow vertex-connection of graphs [article]

Lily Chen, Xueliang Li, Yongtang Shi
2011 arXiv   pre-print
In this paper, we study the computational complexity of vertex-rainbow connection of graphs and prove that computing rvc(G) is NP-Hard.  ...  The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected.  ...  Motivated by the proof of [3] , we consider the computational complexity of rainbow vertex-connection rvc(G) of graphs.  ... 
arXiv:1101.3126v1 fatcat:5yqx5shnp5gidfsfx3wftbgop4

The connectivity of graphs of graphs with self-loops and a given degree sequence

Joel Nishimura
2018 Journal of Complex Networks  
The same classification scheme to characterize degree sequences can be used to prove that, for all degree sequences, loopy graphs are connected by a combination of double and triple edge swaps.  ...  However, while double edge-swaps can transform, for any fixed degree sequence, any two graphs inside the classes of simple graphs, multigraphs, and pseudographs, this is not true for graphs which allow  ...  Categorizing the components of G For any graph G i ∈ G, let V(G i ) be the graphs connected to G i with the maximum number of self-loops.  ... 
doi:10.1093/comnet/cny008 fatcat:7w6u5swk2rd7pfbpxh7kyg4gnu

The Complexity of Connectivity Problems in Forbidden-Transition Graphs and Edge-Colored Graphs [article]

Thomas Bellitto, Shaohua Li, Karolina Okrasa, Marcin Pilipczuk, Manuel Sorge
2020 arXiv   pre-print
We initiate the study of fundamental connectivity problems from the point of view of parameterized complexity, including an in-depth study of tractability with regards to various graph-width parameters  ...  The notion of forbidden-transition graphs allows for a robust generalization of walks in graphs.  ...  As the notion of properly colored walks in edge-colored graphs generalizes walks in directed graphs, the problem in question is more general than finding a cycle of length at least k in a directed graph  ... 
arXiv:2009.12892v1 fatcat:cfdsfxzgtfhjtofbpuwljffaoe

The Complexity of Optimal Design of Temporally Connected Graphs

Eleni C. Akrida, Leszek Gąsieniec, George B. Mertzios, Paul G. Spirakis
2017 Theory of Computing Systems  
We study the design of small cost temporally connected graphs, under various constraints.  ...  We then consider the case in which a designer of temporal graphs can freely choose availability instances for all edges and aims for temporal connectivity with very small cost; the cost is the total number  ...  We wish to give special thanks to the reviewers for the suggestion of the theoretical proof of Theorem 4(a), which now replaces the program code previously used in the proof.  ... 
doi:10.1007/s00224-017-9757-x pmid:32025196 pmcid:PMC6979514 fatcat:yinjlbjl45e4jfjyezykhsupci

On the complexity of partitioning graphs into connected subgraphs

M.E. Dyer, A.M. Frieze
1985 Discrete Applied Mathematics  
This paper is mainly concerned with the computational complexity of determining whether or not the vertices of a graph can be partitioned into equal sized subsets so that each subset induces a particular  ...  type of graph.  ...  (b) If G is 4-edge connected, then G has a connected k-partition for aN k. The question of the complexity of Ek(connected) was left open in the above paper.  ... 
doi:10.1016/0166-218x(85)90008-3 fatcat:l5hj2nqqvngqdecuphgxe6giem

On the connectivity of infinite graphs and 2-complexes

R. Ayala, M.J. Chávez, A. Márquez, A. Quintero
1999 Discrete Mathematics  
This paper contains a study of the connectivity of infinite graphs and 2-complexes. Various connectivity types are defined and relationships among them are given.  ...  In addition new Menger-Whitney type theorems are stated for both graphs and 2-complexes.  ...  Acknowledgements This work was partially supported by the project DGICYT PB96-1374.  ... 
doi:10.1016/s0012-365x(98)00033-8 fatcat:htchfbmxv5bhtcvrhqz7nbzb74

A short proof of a conjecture on the higher connectivity of graph coloring complexes [article]

Alexander Engstrom
2005 arXiv   pre-print
The Hom-complexes were introduced by Lovasz to study topological obstructions to graph colorings.  ...  It was conjectured by Babson and Kozlov, and proved by Cukic and Kozlov, that Hom(G,K_n) is (n-d-2)-connected, where d is the maximal degree of a vertex of G. We give a short proof of the conjecture.  ...  A regular cell complex ∆ is m-connected if there is a family of subcomplexes {∆ i } such that ∆ = ∪∆ i , all of the subcomplexes ∆ i are m-connected, and all of the intersection of several ∆ i 's are (  ... 
arXiv:math/0505460v1 fatcat:ht4asb4ok5aybd53ezrxopvrji
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