NORDHAUS-GADDUM INEQUALITIES FOR THE NUMBER OF CONNECTED INDUCED SUBGRAPHS IN GRAPHS

Let eta(G) be the number of connected induced subgraphs in a graph G, and (G) over bar the complement of G. We prove that eta(G) + eta((G) over bar) is minimum, among all n-vertex graphs, if and only if G has no induced path on four vertices. Since the n-vertex star S-n with maximum degree n - 1 is the unique tree of diameter 2, eta(S-n) + eta((S) over bar (n))is minimum among all n-vertex trees, while the maximum is shown to be achieved only by the tree whose degree sequence is (inverted left parpandicular n/2 inverted right parpendicular, left parpandicular n/2 right parpandicular , 1, ... , 1). Furthermore, we prove that every graph G of n >= 5 order n >= 5 and with maximum eta(G) + eta((G) over bar) must have diameter at most 3, no cut vertex and the property that (G) over bar is also connected. In both cases of trees and graphs that have the same order, we find that if eta(G) is maximum then eta(G) + eta((G) over bar) is minimum. As corollaries to our results, we characterise the unique connected graph G of given order and number of vertices of degree 1, and the unique connected graph G of a given order satisfying vertical bar V (G)vertical bar = vertical bar E(G)vertical bar that minimises eta(G) + eta((G) over bar).