  In this paper we introduce a new technique to analyze families of rankings
focused on the study of structural properties of a new type of graphs. Given a
finite number of elements and a family of rankings of those elements, we say
that two elements compete when they exchange their relative positions in at
least two rankings. This allows us to define an undirected graph by connecting
elements that compete. We call this graph a competitivity graph. We study the
relationship of competitivity graphs with other well-known families of graphs,
such as permutation graphs, comparability graphs and chordal graphs. In
addition to this, we also introduce certain important sets of nodes in a
competitivity graph. For example, nodes that compete among them form a
competitivity set and nodes connected by chains of competitors form a set of
eventual competitors. We analyze hese sets and we show a method to obtain sets
of eventual competitors directly from a family of rankings.
