Games are used to model many instances arising from interaction of more than one computational agent. In program synthesis, existence of strategy is the key in deciding the existence of a program with a given set of specifications.

We will study the Decision-Tree complexity of element distinctness using arbitrary binary gates (an instance of which is comparison gates). Concretely, let $m$ and $n$ be natural numbers with $m>n$.

An undirected graph is chordal if every cycle of length greater than three has a chord: namely, an edge connecting two nonconsecutive vertices on the cycle.  A clique of a graph $G$ is any maximal set of vertices that is complete in $G$.  Let $G$