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How to identify interval graphs in an instant? Where do you know the algorithms?
In graph theory, interval graphs are a rather interesting undirected graph.The definition of these graphs originates from a set of intervals on the real line, and each interval corresponds to a vertex. If the two intervals have intersections, there is an edge between their vertices.Such a structure allows interval diagrams to show great flexibility and effectiveness in various applications.
The interval graph is not only a chord graph, but also a perfect graph, and can be identified in linear time, which means that an algorithm to quickly determine whether a graph is an interval graph exists.
Definition of interval graph
Definitionally, for a family of S_i composed of multiple intervals, we can create a vertex v_i for each interval S_i ; When there are intersections between the two intervals, there will be an edge between the two vertices.This makes the edge set of interval graphs defined as:
E(G) = {(v_i,v_j) | S_i ∩ S_j ≠ ∅}
Features of interval graph
The characteristics of the interval graph are diverse.A graph is an interval graph if and only if it is both a chord graph and a star-free (AT-free).This means there is a unique path in Turikai where no neighbors of the third vertex will participate.It is said that the earliest featureization of interval graphs expanded our understanding of such graphs.
Interestingly, if a graph does not contain quadrilaterals as an induction subgraph, it is also another feature of the interval graph.
Efficient recognition algorithm
To determine whether a given graph G = (V,E) is an interval graph, you can use the complexity of O(|V| + |E|) algorithm.This algorithm identifies the interval graph by finding the sequence of the largest cluster.Although many known algorithms are based on this principle, they can actually identify interval graphs in linear time without using groups.In addition, the algorithm proposed by Booth and Lueker in 1976 utilizes complex PQ tree data structures, while Habib et al. demonstrates a more concise implementation method, using vocabulary breadth-first search (LexBFS) for inspection.
Related Graphics
Based on the characteristics of the AT-free and chord diagrams of the interval graph, we learned that the interval graph also belongs to the category of strong chord diagrams and perfect diagrams.In addition, the complementary diagram of the interval diagram is classified into the category of the comparable diagram.This relationship is critical in explaining why interval graphs become so important in computer science and practical applications.
In terms of application, such as resource allocation problems and scheduling theory, interval graphs provide powerful mathematical tools.
Application of interval graph
The interval map is widely used, ranging from resource allocation to biology food web modeling.Each interval can be regarded as a request for resources, which makes the interval graph a powerful tool in scheduling problems over a specific time period.The best independent set problem can be expressed as finding the best subset of requests, thus not causing conflicts in resources.And the optimal graph shading algorithm can effectively overwrite requests with the least amount of resources.
In genetics and bioinformatics, finding a set of intervals for expression interval graphs can help assemble continuous DNA sequences, and the vigorous development of various applications also makes people look forward to the future of interval graphs.
Think about problems
As the potential application of interval graphs in multiple fields becomes more and more extensive, will these algorithms improve efficiency to solve more practical problems in the future?
