In the
mathematical area of
graph theory, a graph is even-hole-free if it contains no
induced cycle
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 ...
with an even number of
vertices. More precisely, the definition may allow the graph to have induced cycles of length four, or may also disallow them: the latter is referred to as even-cycle-free graphs.
demonstrated that every even-hole-free graph contains a
bisimplicial vertex, which settled a conjecture by Reed.
Recognition
gave the first polynomial time recognition algorithm for even-hole-free graphs, which runs in
time.
later improved this to
.
and improved this to
time.
The best currently known algorithm is given by which runs in
time.
While even-hole-free graphs can be recognized in polynomial time, it is NP-complete to determine whether a graph contains an even hole that includes a specific vertex.
It is unknown whether
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 o ...
and the
maximum independent set problem can be solved in polynomial time on even-hole-free graphs, or whether they are NP-complete.
However the
maximum clique can be found in even-hole-free graphs in polynomial time.
Notes
References
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External links
* {{citation
, contribution-url = http://www.graphclasses.org/classes/gc_547.html
, contribution = Even-hole-free graphs
, url = http://www.graphclasses.org/index.html
, title = Information System on Graph Classes and their Inclusions
Graph families