Edge-connectivity in -path graphs (original) (raw)
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Edge-connectivity in Pk-path graphs
2004
The P k (G)-path graph corresponding to a graph G has for vertices the set of all paths of length k in G. Two vertices are joined by an edge if and only if the intersection of the corresponding paths forms a path of length k − 1 in G, and their union forms either a cycle or a path of length k + 1. Path graphs were introduced by Broersma and Hoede (J. Graph. Theory 13 as a generalization of line graphs, because for k = 1, path graphs are just line graphs. Results on the edge-connectivity of line graphs are given by Chartrand and Stewart (12). The connectivity of P k -path graphs has been studied by Knor and Niepel (Graph Theory 20 (2000) 181), where they proved a necessary and sufficient condition for the P k (G)-path graphs to be disconnected, assuming that G has girth of at least k + 1. Going one step further, we prove in this work that the edge-connectivity of P k (G) is at least (P k (G)) (G) − 1 for a graph G of girth at least k + 1 and minimum degree (G) 2. Furthermore, we show (P k (G)) 2 (G) − 2 provided that (G) 3.
The aim of this paper is to lower bound the connectivity of k-path graphs. From the bounds obtained, we give conditions to guarantee maximum connectivity. Then, it is shown that those maximally connected graphs satisfying the previous conditions are also super-\lambda. While doing so, we derive some properties about the girth and the diameter of path graphs. Finally, the results are extended to path graphs resulting from the iteration of the k-path graph operator.
Results on the edge-connectivity of graphs
Discrete Mathematics, 1974
It is shown that if G is a graph of order p 2 2 such that deg u + deg u 2 p-1 for all pairs u, u of nonadjacent vertices, then the edge-connectivity of G equals the minimum degree of G. Furthermore, if deg u + deg u _> p for all pairs u, u of nonadjacent vertices, then either p is even and G is isomorphic to Kpj2 X K, or every minimum cutset of edges of G consists of the collection of edges incident with a vertex of least degree.
Connected graphs without long paths
Discrete Mathematics, 2008
We determine the maximum number of edges in a connected graph with n vertices if it contains no path with k + 1 vertices. We also determine the extremal graphs. Dedicated to Miklós Simonovits on his 60th birthday Definition. For n ≥ k > 2s > 0 let G n,k,s = (K k−2s ∪ K n−k+s) + K s (see Figure 1). Note that |E(G n,k,s)| = k−s 2 + s(n − k + s) and since k > 2s, G n,k,s contains no P k+1 .
A characterization of graphs without long induced paths
Journal of Graph Theory, 1990
In a connected graph define the k-center as the set of vertices whose distance from any other vertex is at most k. We say that a vertex set S d-dominates G if for every vertex x there is a y E S whose distance from x is at most d.
Edge-connectivityand super edge-connectivity of P2 -path graphs
For a graph G, the P2-path graph, P2(G), has for vertices the set of all paths of length 2 in G. Two vertices are connected when their union is a path or a cycle of length 3. We present lower bounds on the edge-connectivity of the P2 path graph of a connected graph G and give conditions for maximum connectivity. A maximally edge-connected graph is super-lambda if each minimum edge cut is trivial, and it is optimum super-lambda if each minimum nontrivial edge cut consists of all the edges adjacent to one edge. We give conditions on G, for P2(G) to be super-lambda and optimum super-lambda.
Fan-type theorem for path-connectivity
Journal of Graph Theory, 2002
For a connected noncomplete graph G, let m(G ) : minfmax fd G (u), d G vg:d G (u, v) 2g. A well-known theorem of Fan says that every 2-connected noncomplete graph has a cycle of length at least minf|V(G)|, 2m(G)g. In this paper, we prove the following Fan-type theorem: if G is a 3-connected noncomplete graph, then each pair of distinct vertices of G is joined by a path of length at least minf|V(G)|À 1, 2m(G)À 2g. As consequences, we have: (i) if G is a 3-connected noncomplete graph with m(G) > jVGj 2 , then G is Hamilton-connected; (ii) if G is a (s 2)-connected noncomplete graph, where s ! 1 is an integer, then through each path of length s of G there passes a cycle of length ! minf|V(G)|, 2m(G)À sg. Several results known before are generalized and a conjecture of Enomoto, Hirohata, and Ota is proved. ß
On the Average (Edge-)Connectivity of Minimally k-(Edge-)Connected Graphs
ArXiv, 2021
Let G be a graph of order n and let u, v be vertices of G. Let κG(u, v) denote the maximum number of internally disjoint u–v paths in G. Then the average connectivity κ(G) of G, is defined as κ(G) = ∑ {u,v}⊆V (G) κG(u, v)/ ( n 2 ) . If k ≥ 1 is an integer, then G is minimally kconnected if κ(G) = k and κ(G− e) < k for every edge e of G. We say that G is an optimal minimally k-connected graph if G has maximum average connectivity among all minimally k-connected graphs of order n. Based on a recent structure result for minimally 2-connected graphs we conjecture that, for every integer k ≥ 3, if G is an optimal minimally k-connected graph of order n ≥ 2k + 1, then G is bipartite, with the set of vertices of degree k and the set of vertices of degree exceeding k as its partite sets. We show that if this conjecture is true, then κ(G) < 98k for every minimally k-connected graph G. For every k ≥ 3, we describe an infinite family of minimally k-connected graphs whose average connectiv...
Relative lengths of paths and cycles in k-connected graphs
Journal of Combinatorial Theory, Series B, 1982
Let G be a k-connected graph where k) 3. It is shown that if G contains a path L of length I then G also contains a cycle of length at least ((2k-4)/(3k-4)) 1. This result is obtained from a constructive proof that G contains 3k2-7k + 4 cycles which together cover every edge of L at least 2k'-6k + 4 times.