Examples of using Interval graphs in English and their translations into Russian
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They are the connected triangle-free interval graphs.
The connected triangle-free interval graphs are exactly the caterpillar trees.
They are the graphs that are both cographs and interval graphs.
Cohen applied interval graphs to mathematical models of population biology, specifically food webs.
Another important subclass of strongly chordal graphs are interval graphs.
Interval graphs are the intersection graphs of subtrees of path graphs, a special case of trees.
They are the graphs that can be represented as the interval graphs for a set of nested intervals. .
Interval graphs are used to represent resource allocation problems in operations research and scheduling theory.
They are also aspecial case of the circle graphs, something that is not true of interval graphs more generally.
Interval graphs are exactly the graphs that are chordal and that have comparability graph complements.
This property parallels similar relations between pathwidth and interval graphs, and between treewidth and chordal graphs. .
Fuzzy linear interval graphs are constructed in the same way as fuzzy circular interval graphs, but on a line rather than on a circle.
They are a class of co-comparability graphs that contain interval graphs and permutation graphs as subclasses.
Hereditary maximal-clique irreducible graphs include triangle-free graphs, bipartite graphs, and interval graphs.
The split comparability graphs, and therefore also the split interval graphs, can be characterized in terms of a set of three forbidden induced subgraphs.
Unlike interval graphs, however, circular-arc graphs are not always perfect, as the odd chordless cycles C5, C7, etc., are circular-arc graphs. .
Transitive orientability of interval graph complements was proven by Ghouila-Houri(1962); the characterization ofinterval graphs is due to Gilmore& Hoffman 1964.
The interval graphs that have an interval representation in which every two intervals are either disjoint or nested are the trivially perfect graphs. .
Several alternative linear time recognition algorithms are based on breadth-first search orlexicographic breadth-first search rather than on the relation between indifference graphs and interval graphs.
The fact that interval graphs are perfect graphs implies that the number of colors needed, in an optimal arrangement of this type, is the same as the clique number of the interval completion of the net graph. .
A Hamiltonian cycle can be found from a proper interval representation of the graph in time O( n log n){\displaystyle O(n\log n)}, but when the graph itself is given as input, the same problem admits linear-time solution that can be generalized to interval graphs.
For example, the subgraph isomorphism problem is NP-complete on connected proper interval graphs and on connected bipartite permutation graphs, but the induced subgraph isomorphism problem can be solved in polynomial time on these two classes.
Proper interval graphs, the interval graphs formed as intersection graphs of families of intervals in which no interval contains another interval, are claw-free, because four properly intersecting intervals cannot intersect in the pattern of a claw.
Moreover, the induced subtree isomorphism problem(i.e. the induced subgraph isomorphism problem where G2 is restricted to be a tree)can be solved in polynomial time on interval graphs, while the subtree isomorphism problem is NP-complete on proper interval graphs.
Interval graphs are a special case of chordal graphs, and chordal graphs can be represented as intersection graphs of subtrees of a common tree generalizing the way that interval graphs are intersection graphs of subpaths of a path.
The NP-completeness of the achromatic number problem holds also for some special classes of graphs: bipartite graphs, complements of bipartite graphs(that is, graphs having no independent set of more than two vertices),cographs and interval graphs, and even for trees.
Braga, de Souza& Lee(2015a)proved that γm∞ α for all proper interval graphs and the same authors also proved, see Braga, de Souza& Lee(2015b), that there exists a Cayley graph for which the m-eternal domination number does not equal the domination number, contrary to the claim in Goddard, Hededtniemi& Hedetniemi 2005.
A set of nested modules, of which the modular decomposition is an example, can be used to guide the recursive solution of many combinatorial problems on graphs, such as recognizing and transitively orienting comparability graphs, recognizing and finding permutation representations of permutation graphs, recognizing whether a graph is a cograph and finding a certificate of the answer to the question,recognizing interval graphs and finding interval representations for them, defining distance-hereditary graphs( Spinrad, 2003) and for graph drawing Papadoupoulos.
Boxicity, a different way of measuring the complexity of an arbitrary graph in terms of interval graphs Tree-depth, a number that is bounded for a minor-closed graph family if and only if the family excludes a path Degeneracy, a measure of the sparsity of a graph that is at most equal to its path width Graph bandwidth, a different NP-complete optimization problem involving linear layouts of graphs Strahler number, a measure of the complexity of rooted trees defined similarly to pathwidth of unrooted trees Diestel& Kühn 2005.
An interval graph is a graph whose maximal cliques can be ordered in such a way that, for each vertex v, the cliques containing v are consecutive in the ordering.