Cut and pendant vertices and the number of connected induced subgraphs of a graph (original) (raw)

On the maximum number of connected induced subgraphs of a graph

arXiv (Cornell University), 2023

We characterise the structure of those graphs of a given order which maximise the number of connected induced subgraphs for seven different graph classes, each with other prescribed parameters like minimum degree, independence number, vertex cover number, vertex connectivity, edge connectivity, chromatic number, number of bridges, thereby contributing to filling a gap in the literature.

Cut vertex and unicyclic graphs with the maximum number of connected induced subgraphs

arXiv (Cornell University), 2020

Cut vertices are often used as a measure of nodes' importance within a network. They are those nodes whose failure disconnects a graph. Let N(G) be the number of connected induced subgraphs of a graph G. In this work, we investigate the maximum of N(G) where G is a unicyclic graph with n nodes of which c are cut vertices. For all valid n, c, we give a full description of those maximal (that maximise N(.)) unicyclic graphs. It is found that there are generally two maximal unicyclic graphs. For infinitely many values of n, c, however, there is a unique maximal unicyclic graph with n nodes and c cut vertices. In particular, the well-known negative correlation between the number of connected induced subgraphs of trees and the Wiener index (sum of distances) fails for unicyclic graphs with n nodes and c cut vertices: for instance, the maximal unicyclic graph with n = 3, 4 mod 5 nodes and c = n − 5 > 3 cut vertices is different from the unique graph that was shown by Tan et al. [The Wiener index of unicyclic graphs given number of pendant vertices or cut vertices. J. Appl. Math. Comput., 55:1-24, 2017] to minimise the Wiener index. Our main characterisation of maximal unicyclic graphs with respect to the number of connected induced subgraphs also applies to unicyclic graphs with n nodes, c cut vertices and girth at most g > 3, since it is shown that the girth of every maximal graph with n nodes and c cut vertices cannot exceed 4.

Graphs and unicyclic graphs with extremal number of connected induced subgraphs

arXiv (Cornell University), 2018

Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of connected induced subgraphs coincide with those that are known to maximise the Wiener index (the sum of the distances between all unordered pairs of vertices), and vice versa. For every k, we also determine the connected graphs that are extremal with respect to the number of kvertex connected induced subgraphs. We show that, in contrast to the minimum which is uniquely realised by the path, the maximum value is attained by a rich class of connected graphs.

Nordhaus-Gaddum inequalities for the number of connected induced subgraphs in graphs

2020

Let η(G) be the number of connected induced subgraphs in a graph G, and G the complement of G. We prove that η(G)+η(G) is minimum, among all n-vertex graphs, if and only if G has no induced path on four vertices. Since the n-vertex tree S_n with maximum degree n-1 is the unique tree of diameter 2, η(S_n)+η(S_n) is minimum among all n-vertex trees, while the maximum is shown to be achieved only by the tree whose degree sequence is ( n/2, n/2,1,...,1). Furthermore, we prove that every graph G of order n≥ 5 and with maximum η(G)+η(G) must have diameter at most 3, no cut vertex and the property that G is also connected. In both cases of trees and graphs of fixed order, we find that if η(G) is maximum then η(G)+η(G) is minimum. As a corollary to our results, we characterise the unique graph G of given order and number of pendent vertices, and the unique unicyclic graph G of a given order that minimises η(G)+η(G).

The minimum size of graphs satisfying cut conditions

Discrete Applied Mathematics, 2018

A graph G of order n satisfies the cut condition (CC) if there are at least |A| edges between any set A ⊂ V (G), |A| ≤ n/2, and its complement A = V (G) \ A. For even n, G satisfies the even cut condition (ECC), if [A, A] contains at least n/2 edges, for every A ⊂ V (G), |A| = n/2. We investigate here the minimum number of edges in a graph G satisfying CC or ECC. A simple counting argument shows that for both cut conditions |E(G)| ≥ n − 1, and the star K 1,n−1 is extremal. Faudree et al. (1999) conjectured that the extremal graphs with maximum degree ∆(G) < n − 1 satisfying ECC have 3n/2 − O(1) edges. Here we prove the tight bound |E(G)| ≥ 3n/2−3, for every graph G with ∆(G) < n−1 and satisfying CC. If G is 2-connected and satisfies ECC, we prove that |E(G)| ≥ 3n/2 − 2 holds and tight, for every even n. We obtain the weaker bound |E(G)| ≥ 5n/4 − 2, for every graph of order n ≡ 0 (mod 4) with ∆(G) < n − 1 and satisfying ECC; meanwhile we conjecture that |E(G)| ≥ 3n/2 − 4 holds, for every even n.

Induced subgraphs of prescribed size

Journal of Graph Theory, 2003

A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(G) denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open problem of Erdős and McKay we show that every graph G on n vertices for which q(G) ≤ C log n contains an induced subgraph with exactly y edges, for every y between 0 and n δ(C). Our methods enable us also to show that under much weaker assumption, i.e., q(G) ≤ n/14, G still must contain an induced subgraph with exactly y edges, for every y between 0 and e Ω(√ log n) .

On the number of cut-vertices in a graph

International Journal of Mathematics and Mathematical Sciences, 1989

A connected graph with n vertices contains no more than -z(n-2) cutvertices of degree r. All graphs in which the bound is achieved are described. In addition, for graphs of maximum degree three and minimum 6, best possible bounds are obtained for I, 2, 3.

Unavoidable Induced Subgraphs of Large 2-Connected Graphs

arXiv: Combinatorics, 2020

Ramsey proved that for every positive integer nnn, every sufficiently large graph contains an induced KnK_nKn or overlineKn\overline{K}_noverlineKn. Among the many extensions of Ramsey's Theorem there is an analogue for connected graphs: for every positive integer nnn, every sufficiently large connected graph contains an induced KnK_nKn, K1,nK_{1,n}K1,n, or PnP_nPn. In this paper, we establish an analogue for 2-connected graphs. In particular, we prove that for every integer exceeding two, every sufficiently large 2-connected graph contains one of the following as an induced subgraph: KnK_nKn, a subdivision of K2,nK_{2,n}K2,n, a subdivision of K2,nK_{2,n}K2,n with an edge between the two vertices of degree nnn, and a well-defined structure similar to a ladder.