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k-degenerate graphs have also been called k-inductive graphs. [5] The degeneracy of a graph may be computed in linear time by an algorithm that repeatedly removes minimum-degree vertices. [ 6 ] The connected components that are left after all vertices of degree less than k have been (repeatedly) removed are called the k -cores of the graph and ...
A clique-sum of two planar graphs and the Wagner graph, forming a K 5-free graph. In graph theory , Wagner's theorem is a mathematical forbidden graph characterization of planar graphs , named after Klaus Wagner , stating that a finite graph is planar if and only if its minors include neither K 5 (the complete graph on five vertices ) nor K 3,3 ...
k-degenerate graphs have also been called k-inductive graphs. degree 1. The degree of a vertex in a graph is its number of incident edges. [2] The degree of a graph G (or its maximum degree) is the maximum of the degrees of its vertices, often denoted Δ(G); the minimum degree of G is the minimum of its vertex degrees, often denoted δ(G).
Claw-free graphs: Star K 1,3: Induced subgraph Definition Comparability graphs: Induced subgraph Triangle-free graphs: Triangle K 3: Induced subgraph Definition Planar graphs: K 5 and K 3,3: Homeomorphic subgraph Kuratowski's theorem: K 5 and K 3,3: Graph minor Wagner's theorem: Outerplanar graphs: K 4 and K 2,3: Graph minor Diestel (2000), [1 ...
If G is an undirected graph, then the degeneracy of G is the minimum number p such that every subgraph of G contains a vertex of degree p or smaller. A graph with degeneracy p is called p-degenerate. Equivalently, a p-degenerate graph is a graph that can be reduced to the empty graph by repeatedly removing a vertex of degree p or smaller.
There is a notion of degree of outerplanarity. A 1-outerplanar embedding of a graph is the same as an outerplanar embedding. For k > 1 a planar embedding is said to be k-outerplanar if removing the vertices on the outer face results in a (k − 1)-outerplanar embedding. A graph is k-outerplanar if it has a k-outerplanar embedding. [16]
The graph coloring game is a mathematical game related to graph theory. Coloring game problems arose as game-theoretic versions of well-known graph coloring problems. In a coloring game, two players use a given set of colors to construct a coloring of a graph , following specific rules depending on the game we consider.
Wiener's theorem is any of several theorems named after Norbert Wiener: Paley–Wiener theorem; Wiener's 1/ƒ theorem about functions with absolutely convergent Fourier series. Wiener–Ikehara theorem; Wiener–Khinchin theorem; Wiener's tauberian theorem; Wiener–Wintner theorem; See also Wiener's lemma