Digraphs+Theory,+Algorithms+and+Applications_Part11

Digraphs+Theory,+Algorithms+and+Applications_Part11 - 4.5...

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Unformatted text preview: 4.5 Line Digraphs 183 1 2 3 5 4 23 12 H Q 25 34 45 54 Figure 4.4 A digraph H and its line digraph Q = L ( H ). [419] by Beineke and Hemminger. The proof presented here is adapted from [419]. For an n × n-matrix M = [ m ik ], a row i is orthogonal to a row j if ∑ n k =1 m ik m jk = 0. One can give a similar definition of orthogonal columns. Theorem 4.5.1 Let D be a directed pseudograph with vertex set { 1 , 2 ,...,n } and with no parallel arcs and let M = [ m ij ] be its adjacency matrix (i.e., the n × n-matrix such that m ij = 1 , if ij ∈ A ( D ) , and m ij = 0 , otherwise). Then the following assertions are equivalent: (i) D is a line digraph; (ii) there exist two partitions { A i } i ∈ I and { B i } i ∈ I of V ( D ) such that A ( D ) = ∪ i ∈ I A i × B i 4 ; (iii) if vw,uw and ux are arcs of D , then so is vx ; (iv) any two rows of M are either identical or orthogonal; (v) any two columns of M are either identical or orthogonal. Proof: We show the following implications and equivalences: (i) ⇔ (ii), (ii) ⇒ (iii), (iii) ⇒ (iv), (iv) ⇔ (v), (iv) ⇒ (ii). (i) ⇒ (ii). Let D = L ( H ). For each v i ∈ V ( H ), let A i and B i be the sets of in-coming and out-going arcs at v i , respectively. Then the arc set of the subdigraph of D induced by A i ∪ B i equals A i × B i . If ab ∈ A ( D ), then there is an i such that a = v j v i and b = v i v k . Hence, ab ∈ A i × B i . The result follows. (ii) ⇒ (i). Let Q be the directed pseudograph with ordered pairs ( A i ,B i ) as vertices, and with | A j ∩ B i | arcs from ( A i ,B i ) to ( A j ,B j ) for each i and j (including i = j ). Let σ ij be a bijection from A j ∩ B i to this set of arcs (from ( A i ,B i ) to ( A j ,B j )) of Q . Then the function σ defined on V ( D ) by taking σ to be σ ij on A j ∩ B i is a well-defined function of V ( D ) into V ( L ( Q )), since { A j ∩ B i } i,j ∈ I is a partition of V ( D ). Moreover, σ is a bijection since every σ ij is a bijection. Furthermore, it is not difficult to see that σ is an isomorphism from D to L ( Q ) (this is left as Exercise 4.4). 4 Recall that X × Y = { ( x,y ) : x ∈ X,y ∈ Y } . 184 4. Classes of Digraphs (ii) ⇒ (iii). If vw,uw and ux are arcs of D , then there exist i,j such that { u,v } ⊆ A i and { w,x } ⊆ B j . Hence, ( v,x ) ∈ A i × B j and vx ∈ D . (iii) ⇒ (iv). Assume that (iv) does no hold. This means that some rows, say i and j , are neither identical nor orthogonal. Then there exist k,h such that m ik = m jk = 1 and m ih = 1 ,m jh = 0 (or vice versa). Hence, ik,jk,ih are in A ( D ) but jh is not. This contradicts (iii). (iv) ⇔ (v). Both (iv) and (v) are equivalent to the statement: for all i,j,h,k, if m ih = m ik = m jk = 1 , then m jh = 1 ....
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Digraphs+Theory,+Algorithms+and+Applications_Part11 - 4.5...

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