# Parity graph > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Parity_graph > Markdown URL: https://mediated.wiki/source/Parity_graph.md > Source: https://en.wikipedia.org/wiki/Parity_graph > Source revision: 1350851629 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Graph where any two induced paths between nodes both have odd or even lengths}} [[File:Cubic matchstick graph.svg|thumb|A parity graph (the unique smallest [cubic](/source/cubic_graph), [matchstick graph](/source/matchstick_graph)) that is neither [distance-hereditary](/source/Distance-hereditary_graph) nor [bipartite](/source/Bipartite_graph)]] In [graph theory](/source/graph_theory), a '''parity graph''' is a [graph](/source/Graph_(discrete_mathematics)) in which all [induced path](/source/induced_path)s between the same two [vertices](/source/vertex_(graph_theory)) have the same [parity](/source/Parity_(mathematics)): either all paths have odd length, or all have even length.[http://www.graphclasses.org/classes/gc_75.html Parity graphs], Information System on Graph Classes and their Inclusions, retrieved 2016-09-25. This class of graphs was named and first studied by {{harvtxt|Burlet|Uhry|1984}}.{{citation | last1 = Burlet | first1 = M. | last2 = Uhry | first2 = J.-P. | contribution = Parity graphs | doi = 10.1016/S0304-0208(08)72939-6 | mr = 778766 | pages = 253–277 | publisher = North-Holland, Amsterdam | series = North-Holland Math. Stud. | title = Topics on perfect graphs | volume = 88 | year = 1984 | isbn = 978-0-444-86587-8 }}. ==Related classes of graphs== Parity graphs include the [distance-hereditary graph](/source/distance-hereditary_graph)s, in which every two induced paths between the same two vertices have the same length. They also include the [bipartite graph](/source/bipartite_graph)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](/source/line_perfect_graph)s, a generalization of the bipartite graphs. Every parity graph is a [Meyniel graph](/source/Meyniel_graph), 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](/source/perfect_graph)s, parity graphs are also perfect. They are exactly the graphs whose [Cartesian product](/source/Cartesian_product_of_graphs) with a single edge remains perfect.{{citation | last = Jansen | first = Klaus | contribution = A new characterization for parity graphs and a coloring problem with costs | doi = 10.1007/BFb0054326 | mr = 1635464 | pages = 249–260 | publisher = Springer, Berlin | series = Lecture Notes in Comput. Sci. | title = LATIN'98: theoretical informatics (Campinas, 1998) | volume = 1380 | year = 1998| hdl = 11858/00-001M-0000-0014-7BE2-3 | isbn = 978-3-540-64275-6 | hdl-access = free }}. ==Algorithms== A graph is a parity graph if and only if every component of its [split decomposition](/source/split_decomposition) is either a [complete graph](/source/complete_graph) or a [bipartite graph](/source/bipartite_graph).{{citation | last1 = Cicerone | first1 = Serafino | last2 = Di Stefano | first2 = Gabriele | title = On the extension of bipartite to parity graphs | doi = 10.1016/S0166-218X(99)00074-8 | pages = 181–195 | journal = Discrete Appl. Math. | volume = 95 | issue = 1–3 | year = 1999| s2cid = 17260334 | doi-access = free }}. Based on this characterization, it is possible to test whether a given graph is a parity graph in [linear time](/source/linear_time). 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](/source/maximum_independent_set) of a parity graph in [polynomial time](/source/polynomial_time).{{citation | last1 = Cicerone | first1 = Serafino | last2 = Di Stefano | first2 = Gabriele | contribution = On the equivalence in complexity among basic problems on bipartite and parity graphs | doi = 10.1007/3-540-63890-3_38 | mr = 1651043 | pages = 354–363 | publisher = Springer, Berlin | series = Lecture Notes in Comput. Sci. | title = Algorithms and computation (Singapore, 1997) | volume = 1350 | year = 1997 | isbn = 978-3-540-63890-2 }}. ==References== {{reflist}} Category:Graph families Category:Perfect graphs --- Adapted from the Wikipedia article [Parity graph](https://en.wikipedia.org/wiki/Parity_graph) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Parity_graph?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.