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Graph Sandwich Problem
In graph theory and computer science, the graph sandwich problem is a problem of finding a graph that belongs to a particular family of graphs and is "sandwiched" between two other graphs, one of which must be a subgraph and the other of which must be a supergraph of the desired graph. Graph sandwich problems generalize the problem of testing whether a given graph belongs to a family of graphs, and have attracted attention because of their applications and as a natural generalization of recognition problems. Problem statement More precisely, given a vertex set ''V'', a mandatory edge set ''E''1, and a larger edge set ''E''2, a graph ''G'' = (''V'', ''E'') is called a ''sandwich'' graph for the pair ''G''1 = (''V'', ''E''1), ''G''2 = (''V'', ''E''2) if ''E''1 ⊆ ''E'' ⊆ ''E''2. The ''graph sandwich problem'' for property Π is defined as follows:. :Graph Sandwich Problem for Property Π: :Instance: Vertex set ''V'' and edge sets ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical disciplines (including the design and implementation of Computer architecture, hardware and Computer programming, software). Computer science is generally considered an area of research, academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing Vulnerability (computing), security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chordal Graph
In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a ''chord'', which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs. or triangulated graphs.. Chordal graphs are a subset of the perfect graphs. They may be recognized in linear time, and several problems that are hard on other classes of graphs such as graph coloring may be solved in polynomial time when the input is chordal. The treewidth of an arbitrary graph may be characterized by the size of the cliques in the chordal graphs that contain it. Perfect elimination and efficient recognit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparability Graph
In graph theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Comparability graphs have also been called transitively orientable graphs, partially orderable graphs, containment graphs, and divisor graphs. An incomparability graph is an undirected graph that connects pairs of elements that are not comparable to each other in a partial order. Definitions and characterization For any strict partially ordered set , the comparability graph of is the graph of which the vertices are the elements of and the edges are those pairs of elements such that . That is, for a partially ordered set, take the directed acyclic graph, apply transitive closure, and remove orientation. Equivalently, a comparability graph is a graph that has a transitive orientation, an assignment of directions to the edges of the graph (i.e. an orientation of the graph) such that the adjacency relation of the resulting directe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Graph
In the mathematical field of graph theory, a permutation graph is a graph whose vertices represent the elements of a permutation, and whose edges represent pairs of elements that are reversed by the permutation. Permutation graphs may also be defined geometrically, as the intersection graphs of line segments whose endpoints lie on two parallel lines. Different permutations may give rise to the same permutation graph; a given graph has a unique representation (up to permutation symmetry) if it is prime with respect to the modular decomposition. Definition and characterization If \rho = (\sigma_1,\sigma_2,...,\sigma_n) is any permutation of the numbers from 1 to n, then one may define a permutation graph from \sigma in which there are n vertices v_1, v_2, ..., v_n, and in which there is an edge v_i v_j for any two indices i and j for which i\sigma_j. That is, two indices i and j determine an edge in the permutation graph exactly when they determine an inversion in the permutatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chordal Bipartite Graph
In the mathematical area of graph theory, a chordal bipartite graph is a bipartite graph ''B'' = (''X'',''Y'',''E'') in which every cycle of length at least 6 in ''B'' has a ''chord'', i.e., an edge that connects two vertices that are a distance > 1 apart from each other in the cycle. A better name would be weakly chordal and bipartite since chordal bipartite graphs are in general not chordal as the induced cycle of length 4 shows. Characterizations Chordal bipartite graphs have various characterizations in terms of perfect elimination orderings, hypergraphs and matrices. They are closely related to strongly chordal graphs. By definition, chordal bipartite graphs have a forbidden subgraph characterization as the graphs that do not contain any induced cycle of length 3 or of length at least 5 (so-called holes) as an induced subgraph. Thus, a graph ''G'' is chordal bipartite if and only if ''G'' is triangle-free and hole-free. In , two other characterizations are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain Graph
In graph theory, a mixed graph is a graph consisting of a set of vertices , a set of (undirected) edges , and a set of directed edges (or arcs) . Definitions and notation Consider adjacent vertices u,v \in V. A directed edge, called an arc, is an edge with an orientation and can be denoted as \overrightarrow or (u,v) (note that u is the tail and v is the head of the arc). Also, an undirected edge, or edge, is an edge with no orientation and can be denoted as uv or ,v/math>. For the purpose of our application example we will not be considering loops or multiple edges of mixed graphs. A walk in a mixed graph is a sequence v_0,c_1,v_1,c_2,v_2,\dots,c_k,v_k of vertices and edges/arcs such that for all indices i, either c_i=v_v_ is an edge of the graph or c_i=\overrightarrow is an arc of the graph. This walk is a path if it does not repeat any edges, arcs, or vertices, except possibly the first and last vertices. A path is closed if its first and last vertices are the same, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science (journal)
''Theoretical Computer Science'' (TCS) is a computer science journal published by Elsevier, started in 1975 and covering theoretical computer science. The journal publishes 52 issues a year. It is abstracted and indexed by Scopus and the Science Citation Index. According to the Journal Citation Reports, its 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... is 0.827. References Computer science journals Elsevier academic journals Publications established in 1975 {{comp-sci-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Split Graph
In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by , and independently introduced by . A split graph may have more than one partition into a clique and an independent set; for instance, the path is a split graph, the vertices of which can be partitioned in three different ways: #the clique and the independent set #the clique and the independent set #the clique and the independent set Split graphs can be characterized in terms of their forbidden induced subgraphs: a graph is split if and only if no induced subgraph is a cycle on four or five vertices, or a pair of disjoint edges (the complement of a 4-cycle). Relation to other graph families From the definition, split graphs are clearly closed under complementation. Another characterization of split graphs involves complementation: they are chordal graphs the complements of which are also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Threshold Graph
In graph theory, a threshold graph is a graph that can be constructed from a one-vertex graph by repeated applications of the following two operations: # Addition of a single isolated vertex to the graph. # Addition of a single dominating vertex to the graph, i.e. a single vertex that is connected to all other vertices. For example, the graph of the figure is a threshold graph. It can be constructed by beginning with a single-vertex graph (vertex 1), and then adding black vertices as isolated vertices and red vertices as dominating vertices, in the order in which they are numbered. Threshold graphs were first introduced by . A chapter on threshold graphs appears in , and the book is devoted to them. Alternative definitions An equivalent definition is the following: a graph is a threshold graph if there are a real number S and for each vertex v a real vertex weight w(v) such that for any two vertices v,u, uv is an edge if and only if w(u)+w(v)> S. Another equivalent definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Path
In the mathematical area of graph theory, an induced path in an undirected graph is a path that is an induced subgraph of . That is, it is a sequence of vertices in such that each two adjacent vertices in the sequence are connected by an edge in , and each two nonadjacent vertices in the sequence are not connected by any edge in . An induced path is sometimes called a snake, and the problem of finding long induced paths in hypercube graphs is known as the snake-in-the-box problem. Similarly, an induced cycle is a cycle that is an induced subgraph of ; induced cycles are also called chordless cycles or (when the length of the cycle is four or more) holes. An antihole is a hole in the complement of , i.e., an antihole is a complement of a hole. The length of the longest induced path in a graph has sometimes been called the detour number of the graph; for sparse graphs, having bounded detour number is equivalent to having bounded tree-depth. The induced path number of a graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
