The Connectedness Game and thec-Complexity of Certain Graphs

Abstract

When $\mathcal{Z}$ is a finite family of finite sets such that $ \cup \mathcal{Z} \in \mathcal{Z}$, there is an associated game $D( \mathcal{Z} )$ that a certain player can always win by making enough tests, where a test is a special sort of move in the game. The complexity of $\mathcal{Z}$ is defined as the minimum number of tests that suffices to win the game. As a specialization of this notion, there is associated with each connected graph $G = ( {V,E} )$ a game $C( G )$ that involves detecting, in a dynamic setting, the connectedness of a subgraph of G. The number of tests required to win $C( G )$ is called the c-complexity of G. Both the c-complexity and a closely related computational notion are shown to be $O| V |$ when G belongs to a class of graphs that includes all paths and circuits.