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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 ...
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).
A graph is said to be k-generated if for every subgraph H of G, the minimum degree of H is at most k. Incidence chromatic number of k-degenerated graphs G is at most ∆(G) + 2k − 1. Incidence chromatic number of K 4 minor free graphs G is at most ∆(G) + 2 and it forms a tight bound. Incidence chromatic number of a planar graph G is at most ...
For continuous time, the Wiener–Khinchin theorem says that if is a wide-sense-stationary random process whose autocorrelation function (sometimes called autocovariance) defined in terms of statistical expected value = [() ()] <,,, where the asterisk denotes complex conjugate, then there exists a monotone function in the frequency domain < <, or equivalently a non negative Radon measure on ...
A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph K 5 and the complete bipartite graph K 3,3.
Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K 5 (top) or K 3,3 (bottom) subgraphs. Wagner published both theorems in 1937, [1] subsequent to the 1930 publication of Kuratowski's theorem, [2] according to which a graph is planar if and only if it does not contain as a subgraph a subdivision of one of the same two forbidden ...
Characterise word-representable near-triangulations containing the complete graph K 4 (such a characterisation is known for K 4-free planar graphs [126]) Classify graphs with representation number 3, that is, graphs that can be represented using 3 copies of each letter, but cannot be represented using 2 copies of each letter [127]
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]