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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.
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.
In algebraic topology and graph theory, graph homology describes the homology groups of a graph, where the graph is considered as a topological space. It formalizes the idea of the number of "holes" in the graph. It is a special case of a simplicial homology, as a graph is a special case of a simplicial complex. Since a finite graph is a 1 ...
Once either of these rings were held in place (either by pins, with an adhesive, or by hand) any of several provided gearwheels (or rotors)—each having holes for a ballpoint pen—could be spun around the ring to draw geometric shapes. Later, the Super-Spirograph introduced additional shapes such as rings, triangles, and straight bars.
An anti-hole is an induced subgraph of order four whose complement is a cycle; equivalently, it is a hole in the complement graph. This terminology is mainly used in the context of perfect graphs, which are characterized by the strong perfect graph theorem as being the graphs with no odd holes or odd anti-holes. The hole-free graphs are the ...
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.)
According to the Independent, the idea was conjured up by the French company, Bic, who added the holes to prevent people from choking to death if they were to accidentally swallow one.
The art gallery problem or museum problem is a well-studied visibility problem in computational geometry.It originates from the following real-world problem: "In an art gallery, what is the minimum number of guards who together can observe the whole gallery?"