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 Meyniel 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 odd

cycle Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in so ...

of length five or more has at least two

chords Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord ...

(edges connecting non-consecutive vertices of the cycle). The chords may be uncrossed (as shown in the figure) or they may cross each other, as long as there are at least two of them. The Meyniel graphs are named after Henri Meyniel (also known for Meyniel's conjecture), who proved that they 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 in 1976,. long before the proof of the strong perfect graph theorem completely characterized the perfect graphs. The same result was independently discovered by ..

Perfection

The Meyniel graphs are a subclass of the perfect graphs. Every

induced subgraph In the mathematical field of graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges (from the original graph) connecting pairs of vertices in that subset. Definit ...

of a Meyniel graph is another Meyniel graph, and in every Meyniel graph the size of a

maximum clique In the mathematical area of graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph G is an induced subgraph of G that is complet ...

equals the minimum number of colors needed in a

graph coloring In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices ...

. Thus, the Meyniel graphs meet the definition of being a perfect graph, that the clique number equals the chromatic number in every induced subgraph. Meyniel graphs are also called the very strongly perfect graphs, because (as Meyniel conjectured and Hoàng proved) they can be characterized by a property generalizing the defining property of the

strongly perfect graph This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B ...

s: in every induced subgraph of a Meyniel graph, every vertex belongs to an independent set that intersects every

maximal clique In the mathematical area of graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph G is an induced subgraph of G that is comple ...

Related graph classes

The Meyniel graphs contain the

chordal graph In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a ''chord'', which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced c ...

s, the parity graphs, and their subclasses the

interval graph In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. It is the intersection graph of the intervals. ...

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 ...

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, and

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.Meyniel graphs
Information System on Graph Classes and their Inclusions, retrieved 2016-09-25.

Although Meyniel graphs form a very general subclass of the perfect graphs, they do not include all perfect graphs. For instance the house graph (a pentagon with only one chord) is perfect but is not a Meyniel graph.

Algorithms and complexity

Meyniel graphs can be recognized 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 ...

, and several graph optimization problems including

that are

NP-hard In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...

for arbitrary graphs can be solved in polynomial time for Meyniel graphs..

References

{{reflist Graph families Perfect graphs