graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, a parity graph is a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

in which every two

induced path In the mathematical area of graph theory, an induced path in an undirected graph is a path that is an induced subgraph of . That is, it is a sequence of vertices in such that each two adjacent vertices in the sequence are connected by an edge ...

s between the same two vertices have the same parity: either both paths have odd length, or both have even length.Parity graphs
Information System on Graph Classes and their Inclusions, retrieved 2016-09-25. This class of graphs was named and first studied by ..

Related classes of graphs

Parity graphs include the

distance-hereditary graph In graph theory, a branch of discrete mathematics, a distance-hereditary graph (also called a completely separable graph) is a graph in which the distances in any connected induced subgraph are the same as they are in the original graph. Thus, any ...

s, in which every two induced paths between the same two vertices have the same length. They also include the

bipartite graph In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V ar ...

s, which may be characterized analogously as the graphs in which every two paths (not necessarily induced paths) between the same two vertices have the same parity, and the

line perfect graph In graph theory, a line perfect graph is a graph whose line graph is a perfect graph. Equivalently, these are the graphs in which every odd-length simple cycle is a triangle. A graph is line perfect if and only if each of its biconnected c ...

s, a generalization of the bipartite graphs. Every parity graph is a

Meyniel graph In graph theory, a Meyniel graph is a graph in which every odd cycle of length five or more has at least two chords (edges connecting non-consecutive vertices of the cycle). The chords may be uncrossed (as shown in the figure) or they may cross ...

, a graph in which every odd cycle of length five or more has two chords. For, in a parity graph, any long odd cycle can be partitioned into two paths of different parities, neither of which is a single edge, and at least one chord is needed to prevent these from both being induced paths. Then, partitioning the cycle into two paths between the endpoints of this first chord, a second chord is needed to prevent the two paths of this second partition from being induced. Because Meyniel graphs are

perfect graph In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the order of the largest clique of that subgraph ( clique number). Equivalently stated in symbolic terms an arbitrary graph G=(V,E) is per ...

s, parity graphs are also perfect. They are exactly the graphs whose

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ ...

with a single edge remains perfect.

Algorithms

A graph is a parity graph if and only if every component of its

split decomposition In graph theory, a split of an undirected graph is a cut whose cut-set forms a complete bipartite graph. A graph is prime if it has no splits. The splits of a graph can be collected into a tree-like structure called the split decomposition or ...

is either a

complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices ...

or a

. Based on this characterization, it is possible to test whether a given graph is a parity graph in

linear time In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by t ...

. The same characterization also leads to generalizations of some graph optimization algorithms from bipartite graphs to parity graphs. For instance, using the split decomposition, it is possible to find the weighted

maximum independent set In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S of vertices such that for every two vertices in S, there is no edge connecting the two ...

of a parity graph in

polynomial time In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...

References

{{reflist Graph families Perfect graphs