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Partial K-tree
In graph theory, a partial ''k''-tree is a type of graph, defined either as a subgraph of a ''k''-tree or as a graph with treewidth at most ''k''. Many NP-hard combinatorial problems on graphs are solvable in polynomial time when restricted to the partial ''k''-trees, for bounded values of ''k''. Graph minors For any fixed constant ''k'', the partial ''k''-trees are closed under the operation of graph minors, and therefore, by the Robertson–Seymour theorem, this family can be characterized in terms of a finite set of forbidden minors. The partial 1-trees are exactly the forests, and their single forbidden minor is a triangle. For the partial 2-trees the single forbidden minor is the complete graph on four vertices. However, the number of forbidden minors increases for larger values of ''k''. For partial 3-trees there are four forbidden minors: the complete graph on five vertices, the octahedral graph with six vertices, the eight-vertex Wagner graph, and the pentagonal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''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 (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree Decomposition
In graph theory, a tree decomposition is a mapping of a Graph (discrete mathematics), graph into a tree (graph theory), tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the graph. Tree decompositions are also called junction trees, clique trees, or join trees. They play an important role in problems like belief propagation, probabilistic inference, constraint satisfaction, query optimization, and matrix decomposition. The concept of tree decomposition was originally introduced by . Later it was rediscovered by and has since been studied by many other authors. Definition Intuitively, a tree decomposition represents the vertices of a given graph as subtrees of a tree, in such a way that vertices in are adjacent only when the corresponding subtrees intersect. Thus, forms a Glossary of graph theory#Subgraphs, subgraph of the intersection graph of the subtrees. The full intersection graph is a chordal graph. Each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Register Allocation
In compiler optimization, register allocation is the process of assigning local automatic variables and Expression (computer science), expression results to a limited number of processor registers. Register allocation can happen over a basic block (''local register allocation''), over a whole function/Subroutine, procedure (''global register allocation''), or across function boundaries traversed via call-graph (''interprocedural register allocation''). When done per function/procedure the calling convention may require insertion of save/restore around each Call site, call-site. Context Principle {, class="wikitable floatright" , + Different number of general-purpose registers in the most common architectures , - ! Architecture ! scope="col" , 32 bit ! scope="col" , 64 bit , - ! scope="row" , ARM , 15 , 31 , - ! scope="row" , Intel x86 , 8 , 16 , - ! scope="row" , MIPS , 32 , 32 , - ! scope="row" , POWER/PowerPC , 32 , 32 , - ! scope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structured Programming
Structured programming is a programming paradigm aimed at improving the clarity, quality, and development time of a computer program by making specific disciplined use of the structured control flow constructs of selection ( if/then/else) and repetition ( while and for), block structures, and subroutines. It emerged in the late 1950s with the appearance of the ALGOL 58 and ALGOL 60 programming languages, with the latter including support for block structures. Contributing factors to its popularity and widespread acceptance, at first in academia and later among practitioners, include the discovery of what is now known as the structured program theorem in 1966, and the publication of the influential " Go To Statement Considered Harmful" open letter in 1968 by Dutch computer scientist Edsger W. Dijkstra, who coined the term "structured programming". Structured programming is most frequently used with deviations that allow for clearer programs in some particular cases, such as whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compiler
In computing, a compiler is a computer program that Translator (computing), translates computer code written in one programming language (the ''source'' language) into another language (the ''target'' language). The name "compiler" is primarily used for programs that translate source code from a high-level programming language to a lower level language, low-level programming language (e.g. assembly language, object code, or machine code) to create an executable program.Compilers: Principles, Techniques, and Tools by Alfred V. Aho, Ravi Sethi, Jeffrey D. Ullman - Second Edition, 2007 There are many different types of compilers which produce output in different useful forms. A ''cross-compiler'' produces code for a different Central processing unit, CPU or operating system than the one on which the cross-compiler itself runs. A ''bootstrap compiler'' is often a temporary compiler, used for compiling a more permanent or better optimised compiler for a language. Related software ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control-flow Graph
In computer science, a control-flow graph (CFG) is a representation, using graph notation, of all paths that might be traversed through a program during its execution. The control-flow graph was conceived by Frances E. Allen, who noted that Reese T. Prosser used boolean connectivity matrices for flow analysis before. The CFG is essential to many compiler optimizations and static-analysis tools. Definition In a control-flow graph each node in the graph represents a basic block, i.e. a straight-line sequence of code with a single entry point and a single exit point, where no branches or jumps occur within the block. Basic blocks start with jump targets and end with jumps or branch instructions. Directed edges are used to represent jumps in the control flow. There are, in most presentations, two specially designated blocks: the ''entry block'', through which control enters into the flow graph, and the ''exit block'', through which all control flow leaves. Because of its c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biconnected Component
In graph theory, a biconnected component or block (sometimes known as a 2-connected component) is a maximal biconnected subgraph. Any connected graph decomposes into a tree of biconnected components called the block-cut tree of the graph. The blocks are attached to each other at shared vertices called cut vertices or separating vertices or articulation points. Specifically, a cut vertex is any vertex whose removal increases the number of connected components. A block containing at most one cut vertex is called a leaf block, it corresponds to a leaf vertex in the block-cut tree. Algorithms Linear time depth-first search The classic sequential algorithm for computing biconnected components in a connected undirected graph is due to John Hopcroft and Robert Tarjan (1973). It runs in linear time, and is based on depth-first search. This algorithm is also outlined as Problem 22-2 of Introduction to Algorithms (both 2nd and 3rd editions). The idea is to run a depth-first sea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apollonian Network
In combinatorics, 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 graph, planar k-tree, 3-trees, the maximal planar chordal graphs, the uniquely colorable graph, 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. Definition An Apollonian network may be formed, starting from a single triangle graph embedding, embedded in the Euclidean plane, by repeatedly selecting a triangular face of the embedding, adding a new vertex inside the face, and connecting the new vertex to each vertex of the face containing it. In this way, the triangle containing the new vertex is subdivided into three smaller triangles, which may in turn be subdivided in the same way. Examples The complete graphs on three and four ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halin Graph
In graph theory, a Halin graph is a type of planar graph, constructed by connecting the leaves of a tree (graph theory), tree into a cycle. The tree must have at least four vertices, none of which has exactly two neighbors; it should be drawn in the Euclidean plane, plane so none of its edges cross (this is called a planar embedding), and the cycle connects the leaves in their clockwise ordering in this embedding. Thus, the cycle forms the outer face of the Halin graph, with the tree inside it.''Encyclopaedia of Mathematics'', first Supplementary volume, 1988, , p. 281, articl"Halin Graph" and references therein. Halin graphs are named after German mathematician Rudolf Halin, who studied them in 1971.. The cubic graph, cubic Halin graphs – the ones in which each vertex touches exactly three edges – had already been studied over a century earlier by Thomas Kirkman, Kirkman. Halin graphs are polyhedral graphs, meaning that every Halin graph can be used to form the vertices a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outerplanar Graph
In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Outerplanar graphs may be characterized (analogously to Wagner's theorem for planar graphs) by the two forbidden minors and , or by their Colin de Verdière graph invariants. They have Hamiltonian cycles if and only if they are biconnected, in which case the outer face forms the unique Hamiltonian cycle. Every outerplanar graph is 3-colorable, and has degeneracy and treewidth at most 2. The outerplanar graphs are a subset of the planar graphs, the subgraphs of series–parallel graphs, and the circle graphs. The maximal outerplanar graphs, those to which no more edges can be added while preserving outerplanarity, are also chordal graphs and visibility graphs. History Outerplanar graphs were first studied and named by , in connection with the problem of determining the planarity of graphs formed by using a perfect matching to connec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series–parallel Graph
In graph theory, series–parallel graphs are graphs with two distinguished vertices called ''terminals'', formed recursively by two simple composition operations. They can be used to model series and parallel electric circuits. Definition and terminology In this context, the term graph means multigraph. There are several ways to define series–parallel graphs. First definition The following definition basically follows the one used by David Eppstein. A two-terminal graph (TTG) is a graph with two distinguished vertices, ''s'' and ''t'' called ''source'' and ''sink'', respectively. The parallel composition ''Pc = Pc(X,Y)'' of two TTGs ''X'' and ''Y'' is a TTG created from the disjoint union of graphs ''X'' and ''Y'' by merging the sources of ''X'' and ''Y'' to create the source of ''Pc'' and merging the sinks of ''X'' and ''Y'' to create the sink of ''Pc''. The series composition ''Sc = Sc(X,Y)'' of two TTGs ''X'' and ''Y'' is a TTG created from the disjoint union of gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoforest
In graph theory, a pseudoforest is an undirected graphThe kind of undirected graph considered here is often called a multigraph or pseudograph, to distinguish it from a simple graph. in which every Connected component (graph theory), connected component has at most one Cycle (graph theory), cycle. That is, it is a system of Vertex (graph theory), vertices and Edge (graph theory), edges connecting pairs of vertices, such that no two cycles of consecutive edges share any vertex with each other, nor can any two cycles be connected to each other by a path of consecutive edges. A pseudotree is a connected pseudoforest. The names are justified by analogy to the more commonly studied Tree (graph theory), trees and Forest (graph theory), forests. (A tree is a connected graph with no cycles; a forest is a disjoint union of trees.) Gabow and Tarjan. attribute the study of pseudoforests to Dantzig's 1963 book on linear programming, in which pseudoforests arise in the solution of certain F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
