Extremal problems for connected set enumeration

A set of vertices in a graph is connected if it induces a connected subgraph. In this paper we consider the number N(G) of connected sets in a graph G. The problem of determining the value of N(G) is #P-complete in general. We present several sharp upper and lower bounds on N(G) in terms of the order of G and some parameters of G, including chromatic number, stability number, matching number and connectivity and also characterize the extremal graphs at which the bounds are attained. Moreover, we obtain sharp upper bounds on N(G) for restricted classes of graphs such as maximal k-degenerate graphs and maximal outerplanar graphs, with the corresponding extremal graphs. (c) 2023 Elsevier B.V. All rights reserved.