{{short description|Mathematical graph with at least one median per triple of vertices}} {{distinguish|Modular decomposition|Modularity (networks)}} [[File:2d modular lattice.svg|thumb|A modular graph derived from a [[modular lattice]]]] In [[graph theory]], a branch of mathematics, the '''modular graphs''' are [[undirected graph]]s in which every three [[vertex (graph theory)|vertices]] {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} have at least one ''median vertex'' {{math|''m''(''x'', ''y'', ''z'')}} that belongs to [[shortest path]]s between each pair of {{mvar|x}}, {{mvar|y}}, and {{mvar|z}}.<ref name="isgci">[http://www.graphclasses.org/classes/gc_50.html Modular graphs], Information System on Graph Classes and their Inclusions, retrieved 2016-09-30.</ref> Their name comes from the fact that a finite [[Lattice (order)|lattice]] is a [[modular lattice]] if and only if its [[Hasse diagram]] is a modular graph.<ref name="arb">{{citation | last1 = Bandelt | first1 = H.-J. | last2 = Dählmann | first2 = A. | last3 = Schütte | first3 = H. | doi = 10.1016/0166-218X(87)90058-8 | issue = 3 | journal = [[Discrete Applied Mathematics]] | mr = 878021 | pages = 191–215 | title = Absolute retracts of bipartite graphs | volume = 16 | year = 1987| doi-access = free }}.</ref>
It is not possible for a modular graph to contain a cycle of odd length. For, if {{mvar|C}} is a shortest odd cycle in a graph, {{mvar|x}} is a vertex of {{mvar|C}}, and {{mvar|yz}} is the edge of {{mvar|C}} farthest from {{mvar|x}}, there could be no median {{math|''m''(''x'', ''y'', ''z'')}}. In this case, the only vertices on the shortest path {{mvar|yz}} are {{mvar|y}} and {{mvar|z}} themselves. Neither can belong to a shortest path from {{mvar|x}} to the other without shortcutting {{mvar|C}} and creating a shorter odd cycle. Because they have no odd cycles, every modular graph is a [[bipartite graph]].<ref name="isgci"/>
The modular graphs contain as a special case the [[median graph]]s, in which every triple of vertices has a unique median;<ref name="isgci"/> median graphs are related to [[distributive lattice]]s in the same way that modular graphs are related to modular lattices. However, the modular graphs also include other graphs such as the [[complete bipartite graph]]s where the medians are not unique: when the three vertices {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} all belong to one side of the bipartition of a complete bipartite graph, every vertex on the other side is a median.<ref name="arb"/> Every [[chordal bipartite graph]] (a class of graphs that includes the complete bipartite graphs and the bipartite [[distance-hereditary graph]]s) is modular.<ref name="isgci"/>
==References== {{reflist}}
[[Category:Graph families]] [[Category:Bipartite graphs]]