Tournament Graphs

Tournament graphs are used in discrete mathematics to represent a winning vertex in a graph. A tournament is a complete graph in which every pair of vertices are connected by a directed edge. These types of graphs are referred to as tournaments because each of the $$n$$ players competes against the other $$n-1$$ players where ties are not allowed and the winner can be represented on a graph. These graphs are created by assigning every player a vertex and if player $$1$$ beats player $$2$$, then a directed edge can be drawn with the arrow pointing from $$1$$ to $$2.$$ Tournaments graphs create Hamiltonian paths that go through each vertex. The Hamiltonian path theorem states that for every tournament there is a Hamiltonian path for $$n\ge1,$$ for any tournament consisting of $$n$$ vertices in which there is always a sequence of vertices $$v_1,v_2,...,v_n$$ such that $$v_1\rightarrow v_2\rightarrow...\rightarrow v_n.$$
For example,
This tournament consists of four players $$a,b,c$$ and $$d.$$ Notice that in the tournament creates a direct Hamiltonian path going through each vertex where $$A\rightarrow B\rightarrow C\rightarrow D.$$
A widely known tournament graph is based off of the king chicken theorem where $$x$$ is known to be a king chicken if for each opponent $$y,$$ we either have the case  $$x\rightarrow y$$ or there exists another player $$z$$ in which case $$x\rightarrow y\rightarrow z.$$ A king exists if a player can walk to a designated vertex in at most two steps. For example,