Digraphs+Theory,+Algorithms+and+Applications_Part2

Digraphs Theory Alg - 1.2 Digraphs Subdigraphs Neighbours Degrees 3 u z w v x y Figure 1.1 A digraph D i.e u is adjacent to 1 v and v is adjacent

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Unformatted text preview: 1.2 Digraphs, Subdigraphs, Neighbours, Degrees 3 u z w v x y Figure 1.1 A digraph D i.e. u is adjacent to 1 v and v is adjacent to u . If ( u,v ) is an arc, we also say that u dominates v (or v is dominated by u ) and denote it by u → v . We say that a vertex u is incident to an arc a if u is the head or tail of a. We will often denote an arc ( x,y ) by xy . For a pair X,Y of vertex sets of a digraph D , we define ( X,Y ) D = { xy ∈ A ( D ) : x ∈ X,y ∈ Y } , i.e. ( X,Y ) D is the set of arcs with tail in X and head in Y . For example, for the digraph H in Figure 1.2, ( { u,v } , { w,z } ) H = { uw } , ( { w,z } , { u,v } ) H = { wv } , and ( { u,v } , { u,v } ) H = { uv,vu } . u w z v u v w H H z Figure 1.2 A digraph H and a directed pseudograph H . For disjoint subsets X and Y of V ( D ), X → Y means that every vertex of X dominates every vertex of Y , X ⇒ Y stands for ( Y,X ) D = ∅ , and X 7→ Y means that both X → Y and X ⇒ Y hold. For example, in the digraph D of Figure 1.1, u →{ v,w } , { x,y,z }⇒{ u,v,w } and { x,y }7→ z. The above definition of a digraph implies that we allow a digraph to have arcs with the same end-vertices (for example, uv and vu in the digraph H in Figure 1.2), but we do not allow it to contain parallel (also called mul- tiple ) arcs, that is, pairs of arcs with the same tail and the same head, or 1 Some authors use the convention that x is adjacent to y to mean that there is an arc from x to y , rather than just that there is an arc xy or yx in D , as we will do in this book. 4 1. Basic Terminology, Notation and Results loops (i.e. arcs whose head and tail coincide). When parallel arcs and loops are admissible we speak of directed pseudographs ; directed pseudographs without loops are directed multigraphs . In Figure 1.2 the directed pseudo- graph H is obtained from H by appending a loop zz and two parallel arcs from u to w . Clearly, for a directed pseudograph D , A ( D ) and ( X,Y ) D (for every pair X,Y of vertex sets of D ) are multisets (parallel arcs provide re- peated elements). We use the symbol μ D ( x,y ) to denote the number of arcs from a vertex x to a vertex y in a directed pseudograph D . In particular, μ D ( x,y ) = 0 means that there is no arc from x to y . We will sometimes give terminology and notation for digraphs only, but we will provide necessary remarks on their extension to directed pseudographs, unless this is trivial. Below, unless otherwise specified, D = ( V,A ) is a directed pseudograph. For a vertex v in D , we use the following notation: N + D ( v ) = { u ∈ V- v : vu ∈ A } , N- D ( v ) = { w ∈ V- v : wv ∈ A ) } . The sets N + D ( v ) , N- D ( v ) and N D ( v ) = N + D ( v ) ∪ N- D ( v ) are called the out-neighbourhood, in-neighbourhood and neighbourhood of v . We call the vertices in N + D ( v ), N- D ( v ) and N D ( v ) the out-neighbours , in- neighbours and neighbours of v . In Figure 1.2, N + H ( u ) = { v,w } , N- H ( u ) = { v } , N H ( u ) = { v,w...
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Digraphs Theory Alg - 1.2 Digraphs Subdigraphs Neighbours Degrees 3 u z w v x y Figure 1.1 A digraph D i.e u is adjacent to 1 v and v is adjacent

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