Concept
Chordal graph
Related concepts (47)
Windmill graph
In the mathematical field of graph theory, the windmill graph Wd(k,n) is an undirected graph constructed for k ≥ 2 and n ≥ 2 by joining n copies of the complete graph K_k at a shared universal vertex. That is, it is a 1-clique-sum of these complete graphs. It has n(k – 1) + 1 vertices and nk(k − 1)/2 edges, girth 3 (if k > 2), radius 1 and diameter 2. It has vertex connectivity 1 because its central vertex is an articulation point; however, like the complete graphs from which it is formed, it is (k – 1)-edge-connected.
Strongly chordal graph
In the mathematical area of graph theory, an undirected graph G is strongly chordal if it is a chordal graph and every cycle of even length (≥ 6) in G has an odd chord, i.e., an edge that connects two vertices that are an odd distance (>1) apart from each other in the cycle. Strongly chordal graphs have a forbidden subgraph characterization as the graphs that do not contain an induced cycle of length greater than three or an n-sun (n ≥ 3) as an induced subgraph. An n-sun is a chordal graph with 2n vertices, partitioned into two subsets U = {u1, u2,.
Strangulated graph
In graph theoretic mathematics, a strangulated graph is a graph in which deleting the edges of any induced cycle of length greater than three would disconnect the remaining graph. That is, they are the graphs in which every peripheral cycle is a triangle. In a maximal planar graph, or more generally in every polyhedral graph, the peripheral cycles are exactly the faces of a planar embedding of the graph, so a polyhedral graph is strangulated if and only if all the faces are triangles, or equivalently it is maximal planar.
Meyniel graph
In graph theory, a Meyniel graph is a graph in which every odd cycle of length five or more has at least two chords (edges connecting non-consecutive vertices of the cycle). The chords may be uncrossed (as shown in the figure) or they may cross each other, as long as there are at least two of them. The Meyniel graphs are named after Henri Meyniel (also known for Meyniel's conjecture), who proved that they are perfect graphs in 1976, long before the proof of the strong perfect graph theorem completely characterized the perfect graphs.
Perfect graph theorem
In graph theory, the perfect graph theorem of states that an undirected graph is perfect if and only if its complement graph is also perfect. This result had been conjectured by , and it is sometimes called the weak perfect graph theorem to distinguish it from the strong perfect graph theorem characterizing perfect graphs by their forbidden induced subgraphs. A perfect graph is an undirected graph with the property that, in every one of its induced subgraphs, the size of the largest clique equals the minimum number of colors in a coloring of the subgraph.
Hypergraph
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a directed hypergraph is a pair , where is a set of elements called nodes, vertices, points, or elements and is a set of pairs of subsets of . Each of these pairs is called an edge or hyperedge; the vertex subset is known as its tail or domain, and as its head or codomain. The order of a hypergraph is the number of vertices in .
Even-hole-free graph
In the mathematical area of graph theory, a graph is even-hole-free if it contains no induced cycle 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. gave the first polynomial time recognition algorithm for even-hole-free graphs, which runs in time.
Apollonian network
In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.
Indifference graph
In graph theory, a branch of mathematics, an indifference graph is an undirected graph constructed by assigning a real number to each vertex and connecting two vertices by an edge when their numbers are within one unit of each other. Indifference graphs are also the intersection graphs of sets of unit intervals, or of properly nested intervals (intervals none of which contains any other one). Based on these two types of interval representations, these graphs are also called unit interval graphs or proper interval graphs; they form a subclass of the interval graphs.
Path graph
In the mathematical field of graph theory, a path graph (or linear graph) is a graph whose vertices can be listed in the order v_1, v_2, ..., v_n such that the edges are {v_i, v_i+1} where i = 1, 2, ..., n − 1. Equivalently, a path with at least two vertices is connected and has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2. Paths are often important in their role as subgraphs of other graphs, in which case they are called paths in that graph.