Swarthmore College Works Mathematics & Statistics Faculty Works Mathematics & Statistics 2000 Directed Graphs Stephen B. Maurer , '67 Swarthmore College, [email protected] Follow this and additional works at: https://works.swarthmore.edu/fac-math-stat Part of the Discrete Mathematics and Combinatorics Commons Let us know how access to these works benefits ouy Recommended Citation Stephen B. Maurer , '67. (2000). "Directed Graphs". Handbook Of Discrete Mathematics. 526-539. https://works.swarthmore.edu/fac-math-stat/109 This work is brought to you for free by Swarthmore College Libraries' Works. It has been accepted for inclusion in Mathematics & Statistics Faculty Works by an authorized administrator of Works. For more information, please contact [email protected]. 526 Chapters GRAPH THEORY 8.2.1 ATTRIBUTES OF A GRAPH MODEL Definitions: A mathematical representation of a physical or behavioral phenomenon is a corre­ spondence between the parts and processes of that phenomenon and a mathematical system of objects and functions. A model of a physical or behavioral phenomenon is the mathematical object or function assigned to that phenomenon under a mathematical representation. Modeling is the mathematical activity of designing models and comprehensive math­ ematical representations of physical and behavioral phenomena. A graph model is a mathematical representation that involves a graph. Examples: 1. Table 1 gives many examples of graph models. Each example states what the vertices and edges (or arcs) represent and where in the Handbook details on the application can be found. 8.3 DIRECTED GRAPHS Assigning directions to the edges of a graph greatly enhances modeling capability, and is natural whenever order is important, e.g., in a hierarchical structure or a one-way road system. Also, any graph may be viewed as a digraph, by replacing each edge with two directed edges, one in each direction. Many graph problems are best solved as special cases of digraph problems, for instance, finding shortest paths, maximum flows, and connectivity. 8.3.1 DIGRAPH MODELS AND REPRESENTATIONS Most graph terminology applies equally well to digraphs, e.g., subgraph, self-loop, bi­ partite, isomorphic, empty. The definitions below are special to digraphs or take on a somewhat different meaning for digraphs. In context, where it is clear that only di­ graphs are being discussed, “directedness” is often an implicit attribute of an “edge”, “path”, and other terms. Definitions: A directed graph, or digraph, consists of: • a set V, whose elements are called vertices, • a set E, whose elements are called directed edges or arcs, and • an incidence function that assigns to each edge a tail and a head. The tail of an arc is the vertex it leaves, and the head is the vertex it enters. A strict digraph has no self-loops or multi-arcs. The underlying graph of a digraph is the graph obtained from the digraph by replacing every directed edge by an undirected edge. Section 8.3 DIRECTED GRAPHS 527 Table 1 Directory of graph models. subject area vertex attributes and meaning r ^ rGi€r6nc6 and application edge/arc attributes and meaning computer programming vertex labels are program steps §8.1.1 flowcharts edge directions show flow social organization vertices are persons §8.1.1 social networks edges represent interactions civil engineering vertices are road intersections §8.1.1, road networks edges are roads §8.3.1 operations research vertices are activities §8.3.1 scheduling arcs show operational precedence sociology vertices are individuals §8.3.1 hierarchical dominance arcs show who reports to whom computer programming vertices are subprograms §8.3.1 subprogram calling diagram arcs show calling direction ecology vertices are species §8.3.1 food webs arcs show who eats whom operations research vertices are activities to be scheduled §8.3.1, scheduling edges are activity conflicts §8.6.1 genealogy vertices are family members §8.3.1 “family trees” arcs show parenthood set theory vertices are elements §8.3.1 binary relations arcs show relatedness probabilistic analysis vertices are process states §8.3.2 Markov models edges are state transitions traffic control vertices are intersection §8.3.3 assigning one-way streets edges are streets partially ordered sets vertices are elements §8.3.4 Hasse diagrams arcs show covering relation computer engineering vertices are computational nodes §8.4.2 communications networks arcs are communications links operations research vertices are supply and demand nodes §8.4.2 transportation networks arcs are supply lines walking tours vertices are land masses §8.4.3 Seven Bridges of Konigsberg edges are bridges postal delivery routing vertices are street intersections §8.4.3 Chinese Postman Problem edges are streets information theory vertices are binary strings §8.4.4 Gray codes edges are single-bit changes radio broadcasting vertices are broadcast stations §8.6.1 assignment of frequencies edges are potential interference chemistry vertices are chemicals §8.6.1 preventing explosions edges are co-combustibility 528 Chapters GRAPH THEORY vertex attributes and meaning subject area reference and application edge/arc attributes and meaning cartography regions are countries §8.6.4 map-coloring edges are borders highway construction vertices are road intersections §8.7.1 avoiding overcrossings edges are roads electrical network boards vertices are circuit components §8.7.1 avoiding insulation edges are wires VLSI computer chips vertices are circuit components §8.7.4 minimizing layering edges are wires information management vertices are data records §17.1.4 binary search trees edges are decisions computer operating systems vertices are prioritized jobs §17.1.5 priority trees edges are priority relations physical chemistry vertices are atoms §9.3.2 counting isomers edges are molecular bonds network optimization edges are connections §10.1.1 min-cost spanning trees edge-labels are costs bipartite matching parts are people and jobs §10.2.2 personnel assignment edges are job-capabilities network optimization vertices are locations §10.3.1 shortest path edge-labels are distances traveling salesman routing vertices are locations §10.7.1 shortest complete tour edge-labels are distances The out-degree of vertex v, denoted <5+(u), is the number of arcs with tail at v. The in-degree of vertex v, denoted <5“(u), is the number of arcs with head at v. A digraph D is transitive if whenever it contains an arc from utov and an arc from v to w, it also contains an arc from u to w. The adjacency matrix Ad of a digraph D is Ad = [ffltj]) where Cy = number of arcs from Vi to vj. The incidence matrix Md of a digraph D with no self-loops is Md = [6»j], where -t-1, if Ui is the tail of ej but not the head {-1, if Vi is the head of Cj but not the tail 0, otherwise. There is no standard convention for self-loops. Facts: 1. Strict-digraph terminology: In a context focusing primarily on strict digraphs, there is often a different terminological convention: • “digraph” refers to a strict digraph; • a directed graph with multi-arcs is called a multidigraph; • a directed graph with self-loops is called a pseudodigraph; • an arc with tail u and head v is designated uv. Section 8.3 DIRECTED GRAPHS 529 2. Alternative “path” terminology: There is an alternative convention in which a (directed) “path” may use vertices and arcs more than once, but an “elementary path” does not repeat arcs, and a “simple path” does not repeat vertices (and, hence, does not repeat arcs either). See §8.3.2. 3. The incidence structure of a digraph is frequently represented by an arc list, in which each arc is represented by an ordered pair uv, where u is its tail and v is its head. For each arc with tail u and head v, there is a separate entry, so that uv occurs as often as the number of such arcs. A list of the isolated vertices plus such an arc list completely specifies a digraph. 4. Another common specification of a digraph is the lists-of-neighbors representation. For each vertex u, there is a corresponding row, which has as an entry the head of each arc whose tail is u. Thus a vertex v occurs in that row as many times as there are arcs from u to V. 5. The incidence matrix is another common way to represent a digraph. Since all but one or two of the entries in every column are zero, the incidence matrix is a highly inefficient form of representation. 6. The adjacency matrix is also a common way to specify a digraph in some contexts when there is no reason to identify the arcs by name. 7. A digraph can be represented by a 2 x \E\ incidence table in which the tail and head of each arc e appear in column e. Direction on an arc can be indicated by a convention as to whether tail or head appears in the first row, which requires swapping the two column entries if the direction is changed. Alternatively, direction can be indicated by marking one of the two entries in each column as the head, and then moving the marker if the direction changes. 8. A row-sum in a directed adjacency matrix equals the out-degree of the corresponding vertex. A column-sum equals the in-degree. 9. In any digraph, the sum of the in-degrees, the sum of the out-degrees, and the number of edges are all equal to each other; i.e., <^^(^) = l-^l- Examples; 1. The following arc list, incidence table, list-of-neighbors, and adjacency matrix all represent the digraph G. ^5 incidence table: arc list: ei 62 63 64 65 66 67 uv, vv, vw, XW, XW, UX, XU U V V X X U X VV w W w X u 530 Chapters GRAPH THEORY u V w X ' u : v,x u /O 1 0 1\ V : v,w 0 1 1 0 lists-of-neighbors; adjacency matrix: ^V w : 0 ► 0 0 0 0 X 0 2 0/ .