The Generalized Connectivity of Generalized Petersen Graph

2021 ◽  
Author(s):  
Yuan Si ◽  
Ping Li ◽  
Yuzhi Xiao ◽  
Jinxia Liang
Keyword(s):  
Steiner Trees ◽  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Edge Disjoint ◽  
Generalized Connectivity ◽  
Vertex Set

For a vertex set [Formula: see text] of [Formula: see text], we use [Formula: see text] to denote the maximum number of edge-disjoint Steiner trees of [Formula: see text] such that any two of such trees intersect in [Formula: see text]. The generalized [Formula: see text]-connectivity of [Formula: see text] is defined as [Formula: see text]. We get that for any generalized Petersen graph [Formula: see text] with [Formula: see text], [Formula: see text] when [Formula: see text]. We give the values of [Formula: see text] for Petersen graph [Formula: see text], where [Formula: see text], and the values of [Formula: see text] for generalized Petersen graph [Formula: see text], where [Formula: see text] and [Formula: see text].

The groups of the generalized Petersen graphs

1971 ◽  
Vol 70 (2) ◽  
pp. 211-218 ◽  
Author(s):  
Roberto Frucht ◽  
Jack E. Graver ◽  
Mark E. Watkins
Keyword(s):  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Generalized Petersen Graphs ◽  
Trivalent Graph ◽  
Vertex Set ◽  
Petersen Graphs ◽  
Image Position

1.Introduction. For integersnandkwith 2 ≤ 2k <n, thegeneralized Petersen graph G(n, k)has been defined in (8) to have vertex-setand edge-setE(G(n, k))to consist of all edges of the formwhereiis an integer.All subscripts in this paper are to be read modulo n, where the particular value ofnwill be clear from the context. ThusG(n, k)is always a trivalent graph of order 2n, andG(5, 2) is the well known Petersen graph. (The subclass of these graphs withnandkrelatively prime was first considered by Coxeter ((2), p. 417ff.).)

scholarly journals On Structural Parameterizations of the Edge Disjoint Paths Problem

Algorithmica ◽  
2021 ◽  
Author(s):  
Robert Ganian ◽  
Sebastian Ordyniak ◽  
M. S. Ramanujan
Keyword(s):  
Polynomial Time ◽  
Disjoint Paths ◽  
Structural Parameter ◽  
Input Graph ◽  
Feedback Vertex Set ◽  
Fpt Algorithms ◽  
Edge Disjoint ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Terminal Pair

AbstractIn this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain -hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the [1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

On complete subgraphs of different orders

1976 ◽  
Vol 79 (1) ◽  
pp. 19-24 ◽  
Author(s):  
Béla Bollobás
Keyword(s):  
Intersection Graph ◽  
Edge Disjoint ◽  
Vertex Set

Let S be a set and let {X1, …, Xn} = be a family of distinct subsets of S. The intersection graph Ω() of has vertex set {X1, …, Xn} and XiXj (i ≠ j) is an edge of Ω() if and only if Xi ∩ Xi ≠ Ø (c.f. (6)). It is easily seen, (7), that every graph is an intersection graph, in other words every graph can be represented by subsets ofa set. Moreover, it was proved by Erdös, Goodman and Pósa (5) that if a graph has n ≥ 4 vertices then one can find a representing set with at most [n2/4] elements. This assertion is an immediate consequence of the result, (5), that the edges of a graph with n ≥ 1 vertices can be covered with at most [n2/4] edge disjoint triangles and edges. We say that a graph G is covered with the subgraphs G1, …, Gk, if every edge of G is in at least one Gi. One of the aims of this note is to prove an extension of this result, pro-posed by Erdös (4).

scholarly journals A Note on the Linear Cycle Cover Conjecture of Gyárfás and Sárközy

10.37236/7329 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Beka Ergemlidze ◽  
Ervin Győri ◽  
Abhishek Methuku
Keyword(s):  
Independent Set ◽  
Uniform Hypergraph ◽  
Edge Disjoint ◽  
Cycle Cover ◽  
Vertex Set ◽  
Cyclic Sequence

A linear cycle in a $3$-uniform hypergraph $H$ is a cyclic sequence of hyperedges such that any two consecutive hyperedges intersect in exactly one element and non-consecutive hyperedges are disjoint. Let $\alpha(H)$ denote the size of a largest independent set of $H$.We show that the vertex set of every $3$-uniform hypergraph $H$ can be covered by at most $\alpha(H)$ edge-disjoint linear cycles (where we accept a vertex and a hyperedge as a linear cycle), proving a weaker version of a conjecture of Gyárfás and Sárközy.

Edge disjoint Steiner trees in graphs without large bridges

2009 ◽  
Vol 62 (2) ◽  
pp. 188-198 ◽  
Author(s):  
Matthias Kriesell
Keyword(s):  
Steiner Trees ◽  
Edge Disjoint

scholarly journals REPARTITORS, SELECTORS AND SUPERSELECTORS

2006 ◽  
Vol 07 (03) ◽  
pp. 391-415 ◽  
Author(s):  
FRÉDÉRIC HAVET
Keyword(s):  
Lower Bounds ◽  
Disjoint Paths ◽  
Edge Disjoint ◽  
Minimum Number ◽  
Vertex Set ◽  
Optimal Networks

An (n, p, f)-network G is a graph (V, E) where the vertex set V is partitioned into four subsets [Formula: see text] and [Formula: see text] called respectively the priorities, the ordinary inputs, the outputs and the switches, satisfying the following constraints: there are p priorities, n - p ordinary inputs and n + f outputs; each priority, each ordinary input and each output is connected to exactly one switch; switches have degree at most 4. An (n, p, f)-network is an (n, p, f)-repartitor if for any disjoint subsets [Formula: see text] and [Formula: see text] of [Formula: see text] with [Formula: see text] and [Formula: see text], there exist in G, n edge-disjoint paths, p of them from [Formula: see text] to [Formula: see text] and the n - p others joining [Formula: see text] to [Formula: see text]. The problem is to determine the minimum number R(n, p, f) of switches of an (n, p, f)-repartitor and to construct a repartitor with the smallest number of switches. In this paper, we show how to build general repartitors from (n, 0, f)-repartitors also called (n, n + f)-selectors. We then consrtuct selectors using more powerful networks called superselectors. An (n, 0, 0)-network is an n-superselector if for any subsets [Formula: see text] and [Formula: see text] with [Formula: see text], there exist in G, [Formula: see text] edge-disjoint paths joining [Formula: see text] to [Formula: see text]. We show that the minimum number of switches of an n-superselector S+ (n) is at most 17n + O(log(n)). We then deduce that [Formula: see text] if [Formula: see text], R(n, p, f) ≤ 18n + 34f + O( log (n + f)), if [Formula: see text] and [Formula: see text] if [Formula: see text]. Finally, we give lower bounds for R(n, 0, f) and S+ (n) and show optimal networks for small value of n.

scholarly journals The K -Size Edge Metric Dimension of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Tanveer Iqbal ◽  
Muhammad Naeem Azhar ◽  
Syed Ahtsham Ul Haq Bokhary
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Petersen Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Generalized Petersen Graph

In this paper, a new concept k -size edge resolving set for a connected graph G in the context of resolvability of graphs is defined. Some properties and realizable results on k -size edge resolvability of graphs are studied. The existence of this new parameter in different graphs is investigated, and the k -size edge metric dimension of path, cycle, and complete bipartite graph is computed. It is shown that these families have unbounded k -size edge metric dimension. Furthermore, the k-size edge metric dimension of the graphs Pm □ Pn, Pm □ Cn for m, n ≥ 3 and the generalized Petersen graph is determined. It is shown that these families of graphs have constant k -size edge metric dimension.

scholarly journals Double covers of graphs

1976 ◽  
Vol 14 (2) ◽  
pp. 233-248 ◽  
Author(s):  
Derek A. Waller
Keyword(s):  
Theoretical Approach ◽  
Upper Bound ◽  
Double Cover ◽  
Petersen Graph ◽  
Classification Theorem ◽  
Finite Graphs ◽  
Vertex Set ◽  
Double Covers

A projection morphism ρ: G1 → G2 of finite graphs maps the vertex-set of G1 onto the vertex-set of G2, and preserves adjacency. As an example, if each vertex v of the dodecahedron graph D is identified with its unique antipodal vertex v¯ (which has distance 5 from v) then this induces an identification of antipodal pairs of edges, and gives a (2:1)-projection p: D → P where P is the Petersen graph.In this paper a category-theoretical approach to graphs is used to define and study such double cover projections. An upper bound is found for the number of distinct double covers ρ: G1 → G2 for a given graph G2. A classification theorem for double cover projections is obtained, and it is shown that the n–dimensional octahedron graph K2,2,…,2 plays the role of universal object.

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