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Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing. The non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps (i.e. a non-orientable surface of (non-orientable) genus n). (This number is also called the demigenus.)
The even-hole-free graphs are the graphs containing no induced cycles with an even number of vertices. The trivially perfect graphs are the graphs that have neither an induced path of length three nor an induced cycle of length four. By the strong perfect graph theorem, the perfect graphs are the graphs with no odd hole and no odd antihole.
A chordal graph, a special type of perfect graph, has no holes of any size greater than three. The girth of a graph is the length of its shortest cycle; this cycle is necessarily chordless. Cages are defined as the smallest regular graphs with given combinations of degree and girth.
The edges and vertices of these six regions form Tietze's graph, which is a dual graph on this surface for the six-vertex complete graph but cannot be drawn without crossings on a plane. Another family of graphs that can be embedded on the Möbius strip, but not on the plane, are the Möbius ladders , the boundaries of subdivisions of the ...
The graph of the 3-3 duoprism (the line graph of ,) is perfect. Here it is colored with three colors, with one of its 3-vertex maximum cliques highlighted. In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the ...
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. [ 3 ] It is unknown whether graph coloring and the maximum independent set problem can be solved in polynomial time on even-hole-free graphs, or whether they are NP-complete.
A toroidal graph that cannot be embedded in a plane is said to have genus 1. The Heawood graph, the complete graph K 7 (and hence K 5 and K 6), the Petersen graph (and hence the complete bipartite graph K 3,3, since the Petersen graph contains a subdivision of it), one of the Blanuša snarks, [1] and all Möbius ladders are toroidal.
A "k-dimensional hole" is a k-dimensional cycle that is not a boundary of a (k+1)-dimensional object. The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional simplicial complexes: b 0 is the number of connected components; b 1 is the number of one-dimensional or "circular" holes;
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