Comparing rankings by means of competitivity graphs: structural properties and computation

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 these sets and we show a method to obtain sets of eventual competitors directly from a family of rankings.