Bidirected graph From Wikipedia, the free encyclopedia Contents 1 Bidirected graph 1 1.1 Other meanings ............................................ 1 1.2 See also ................................................ 2 1.3 References ............................................... 2 2 Bipartite double cover 3 2.1 Construction .............................................. 3 2.2 Examples ............................................... 3 2.3 Matrix interpretation ......................................... 4 2.4 Properties ............................................... 4 2.5 Other double covers .......................................... 4 2.6 See also ................................................ 5 2.7 Notes ................................................. 5 2.8 References ............................................... 5 2.9 External links ............................................. 6 3 Complex question 7 3.1 Implication by question ........................................ 7 3.2 Complex question fallacy ....................................... 7 3.2.1 Similar questions and fallacies ................................ 8 3.3 Notes ................................................. 8 4 Directed graph 10 4.1 Basic terminology ........................................... 11 4.2 Indegree and outdegree ........................................ 11 4.3 Degree sequence ............................................ 12 4.4 Digraph connectivity .......................................... 12 4.5 Classes of digraphs .......................................... 13 4.6 See also ................................................ 14 4.7 Notes ................................................. 15 4.8 References ............................................... 15 5 Downward entailing 16 5.1 Strawson-DE ............................................. 16 i ii CONTENTS 5.2 See also ................................................ 17 5.3 References ............................................... 17 6 Entailment (pragmatics) 18 6.1 Types of entailment .......................................... 18 6.2 See also ................................................ 18 6.3 References ............................................... 18 6.4 Further reading ............................................ 18 7 Fallacy 19 7.1 Formal fallacy ............................................. 19 7.1.1 Common examples ...................................... 19 7.2 Aristotle's Fallacies .......................................... 19 7.3 Whately's grouping of fallacies .................................... 20 7.4 Intentional fallacies .......................................... 20 7.5 Deductive fallacy ........................................... 20 7.6 Paul Meehl's Fallacies ......................................... 20 7.7 Fallacies of Measurement ....................................... 21 7.8 Other systems of classification .................................... 22 7.9 Assessment of Fallacies - Pragmatic Theory ............................. 22 7.10 See also ................................................ 22 7.11 References .............................................. 23 7.12 Further reading ............................................ 24 7.13 External links ............................................. 25 8 Fixed point (mathematics) 26 8.1 Attractive fixed points ......................................... 27 8.2 Applications .............................................. 28 8.3 Topological fixed point property .................................... 28 8.4 Generalization to partial orders: prefixpoint and postfixpoint ..................... 29 8.5 See also ................................................ 29 8.6 Notes ................................................. 29 8.7 External links ............................................. 30 9 Graph isomorphism 31 9.1 Variations ............................................... 31 9.2 Motivation ............................................... 31 9.3 Whitney theorem ........................................... 32 9.4 Recognition of graph isomorphism .................................. 32 9.5 See also ................................................ 33 9.6 Notes ................................................. 33 9.7 References ............................................... 33 CONTENTS iii 10 Graph theory 34 10.1 Definitions ............................................... 35 10.1.1 Graph ............................................. 35 10.2 Applications .............................................. 35 10.3 History ................................................. 37 10.4 Graph drawing ............................................. 38 10.5 Graph-theoretic data structures .................................... 38 10.6 Problems in graph theory ....................................... 39 10.6.1 Enumeration ......................................... 39 10.6.2 Subgraphs, induced subgraphs, and minors .......................... 39 10.6.3 Graph coloring ........................................ 39 10.6.4 Subsumption and unification ................................. 40 10.6.5 Route problems ........................................ 40 10.6.6 Network flow ......................................... 40 10.6.7 Visibility problems ...................................... 40 10.6.8 Covering problems ...................................... 40 10.6.9 Decomposition problems ................................... 41 10.6.10 Graph classes ......................................... 41 10.7 See also ................................................ 41 10.7.1 Related topics ......................................... 41 10.7.2 Algorithms .......................................... 42 10.7.3 Subareas ........................................... 43 10.7.4 Related areas of mathematics ................................. 43 10.7.5 Generalizations ........................................ 43 10.7.6 Prominent graph theorists ................................... 43 10.8 Notes ................................................. 44 10.9 References ............................................... 45 10.10External links ............................................. 45 10.10.1 Online textbooks ....................................... 45 11 Implication graph 46 11.1 Applications .............................................. 47 11.2 References ............................................... 47 12 Implicational hierarchy 48 12.1 Phonology ............................................... 48 12.2 Morphology .............................................. 48 12.3 Syntax ................................................. 48 12.4 Bibliography .............................................. 49 13 Implicational propositional calculus 50 13.1 Virtual completeness as an operator .................................. 50 iv CONTENTS 13.2 Axiom system ............................................. 50 13.3 Basic properties of derivation ..................................... 51 13.4 Completeness ............................................. 51 13.4.1 Proof ............................................. 51 13.5 The Bernays–Tarski axiom system .................................. 53 13.6 Testing whether a formula of the implicational propositional calculus is a tautology ......... 53 13.7 Adding an axiom schema ....................................... 54 13.8 An alternative axiomatization ..................................... 54 13.9 See also ................................................ 56 13.10References ............................................... 56 14 Implicature 57 14.1 Types of implicature .......................................... 57 14.1.1 Conversational implicature .................................. 57 14.1.2 Conventional implicature ................................... 58 14.2 Implicature vs entailment ....................................... 58 14.3 See also ................................................ 58 14.4 References ............................................... 58 14.5 Bibliography .............................................. 59 14.6 Further reading ............................................ 59 14.7 External links ............................................. 59 15 Implicit 60 15.1 Mathematics ............................................. 60 15.2 Computer science ........................................... 60 15.3 Other uses ............................................... 60 15.4 See also ................................................ 60 16 Informal fallacy 61 16.1 Formal deductive fallacies and informal fallacies ........................... 61 16.2 See also ................................................ 61 16.3 References ............................................... 62 16.4 Further reading ............................................ 62 16.5 External links ............................................. 62 17 Involution (mathematics) 63 17.1 General properties ........................................... 63 17.2 Involution throughout the fields of mathematics ............................ 64 17.2.1 Euclidean geometry ...................................... 64 17.2.2 Projective geometry ...................................... 64 17.2.3 Linear algebra ......................................... 64 17.2.4 Quaternion algebra, groups, semigroups ........................... 64 17.2.5 Ring theory .......................................... 65 CONTENTS v 17.2.6 Group theory ......................................... 65 17.2.7 Mathematical logic ...................................... 65 17.2.8 Computer science .....................................