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3.4   Resultand AnalysisNowwe explore some domination related results on digraphs analogous to those ofundirected graphs. First we look at some common b?unds  for ?(D). One of the earliest b?unds for the domination number forany undirected graph was proposed by Ore. Theorem3.41(Ore) For any graph G without isolates, ?(G) ?  , wheren is the number of vertices.

This result does not hold for directedgraphs; ? counter example is the digraph K1,n,n ? 2, with its arcs directed from the end vertices towards the centralvertex. The general bound which holds for digraphs is not very good for ?majority of digraphs. We assume our digraphs to be those whose underlyinggraphs are connected.

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 Observation3.42   For any digraph’with n vertices, ?(D) ? n -1.Thisbound is sharp because the domination number of the digraph ?1,n forn ? 2 with its arcs directedfrom the end vertices towards the central vertex is n. Since very few graphs agreewith this bound we find other bounds which are tighter for ? significant number of digraphs.

 Theorem3.43For any digraph D on n vertices,  ? ?(D) ? n –  (D), wh???    (D) denotesthe maximumoutdegree.Proof.  For the upper bound weform ? dominating set of D by including the vertex ? of maximum outdegree andall the other vertices in the digraph which are not dominated by v. This set isclearly ? dominating set and has cardinality n –  (D). Note that any vertex in D can dominate at most 1 +  (D) vertices.In ? minimum dominating set S of D thereare ?(D) vertices, so they can dominate at most ?(D)(1 +  (D)) vertices. Since S is dominating thisnumber has to be at least n.

Thuswe get the lower bound.?? get another bound we look for certaincharacteristics in ? digraph.Observation 3.

44 For any digraph D on n vertices, which has ? hamiltonian circuit, ?(D) ?  Proof.LetD contain ? hamiltonian circuit?. ?? dominate the vertices of D it suffices to dominate the cycle ?. We know that the domination numberof ? circuit is bounded above by and so the same holds for thedigraph D.

Theorem3.45For ? strongly connected digraph D onn vertices, ?(D) ?  Inaddition to ?(D) we introduce some domination related parameters in digraphs,in particular, the irredundancenumber, the u???r irredundancenumber and the u???? dominationnumber, analogous to those for undirected graphs. Recall that ? set S ?V(D) of ? digraph D is? dominating set if for all v S, vis ? successor of some vertex in S. ? dominating set S is? minimal dominating set if forevery v S,Ov-OS-v  If u  O? – OS – ? , then u will be called ? private outneighbor (??n) of ? withrespect to S. See 38 for another characterization of minimal dominating setsin digraphs. Let ?(D), the upper domination??????, denote the maximum cardinality of ? minimal dominating set. As in theundirected case, we define an irredundantset S ?V(D) to be ? set such that every ?  S has ? private outneighbor. The irredundance number ir(D) and the ????? irredundance number IR(D) are,respectively, the minimum and maximum cardinalities of ? maximal irredundantset.

The notion of ?solution also yields parameters which are new to the field of digraphs. Leti(D) and (D) denote respectively the minimum and maximumcardinalitiesof an independent dominating set. It must be pointed out that not ?ll digraphshave independent dominating sets. As these are special cases of solutions,these exist in digraphs which admit at least one solution. It must be mentionedhere that due to the concepts defined above the following string ofinequalities hold for any digraph D with ? solution,r(D) ? ?(D) ? i(D) ? ?(D) ? ?(D) ? IR(D).             Researchersinterested in domination theory for undirected graphs are quite  familiar with the corresponding inequalitychain. This chain raises some interesting  questions about the structural properties ofdigraphs D (having ? solution),for which 1.?(D) = i(D),2.

ir(D) = ?(D), ?.?(D) = ?(D) = IR(D), or 4. i(D) ? ?(D).  Thefollowing theorem is an interesting result for transitive digraphs.Theorem3.46   In ? transitive digraphD, we have ?(D) = i(D) =?(D)= ?(D) = IR(D).Proof.

  Note that if D is ? transitive digraph so is itsreversal D-1. It isthen known that ? solution exists in D.Moreover, from Berge’s theorem, we see that in D, every minimal absorbant set is independent and the kernel isunique. This implies that ?(D) = I(D)= ?(D)= ?(D). ?? show ? (D) = IR(D), suppose that S is an irredundant set with ?S? = IR(D). We will call such ? set an IR-set. Amongst all IR-sets let Scontain the minimum number of arcs in it.

If S has no arcs, then certainly S isindependent and ? (D) ? IR(D) implying? (D) = IR(D). So suppose that< S > contains an arc (?, ?). Since S is irredundant y must have ?private outneighbor ?1 S. But D is ? transitive digraph, so (? ,?1) must be an arc,contradicting that y1 is ? private neighbor of x. Hence S isindependent and the result follows.

3.5.Applications 3.51. GameTheory (Von Neumann,Morgenstern ) Supposethat ? players, denoted by (1),(2),…,(n) can discuss together toselect ? point x from ? set ? (the “situations”).

Ifplayer (i) prefers situation ? to situation b, we shall write ? ?Ib. The individual preferences might not be compatible, and consequently it isnecessary to introduce the concept of effective preference. Thesituation ? is said to be effectively preferred to b, or ? >b, if there is ? set of players who prefer ? to b and who are ?lltogether capable of enforcing their preference for ?.

However, effectivepreference is not transitive; i.?., ?> b and b > ? does notnecessarily imply that ? > ?.Consider the digraph D = (V, ?) where O(x)denotes the set of situations effectively preferred to x. Let S be ? kernel of D.Von Neumann and Morgen-stern suggested that the selection be confined tothe elements of S.

Since S is independent, n? situation in S is effectivelypreferred to any other situation in S. Since S is absorbant, for everysituation x  there is ? situation in S that is effectivelypreferred to x, so that x can be immediately discarded. 3.

52. Problemin Logic (Berge,Rao 06) Letus consider ? set of properties P = {?1,p2} and ?set of theorems of the type: “property ?i implies property ?j’. These theorems can be represented by ? directed graph D = (V, ?) with vertexset P, where (?i, ?j) is an ??? if and only if itfollows from one or more of the existing theorems that ?i implies ?jSuppose we want to show that n? arc of the complementary graph is good to represent ? trueimplication of that kind: more precisely, with each ??? (?, q) with ? ? q and we assign ? student who has to findan example where ? is fulfilled but not q (i.?.

, ? counter-example to thestatement that p implies q). In10 they determined the minimum number of students needed to show that ?ll thepossible (pairwise) implications are already represented in the di-graph D. Itwas found that this number corresponded to the cardinality of the unique kernelof the transitive digraph under study. 3.

53   Facility LocationLet D = (V, ?) be ? digraph where thevertices represent “locations” and there is an ??? from location ?to location ? if location ? can be “reached?’ fromlocation ?. Assume that each “location” has ? weightassociated with it which represents some parameter pertinent to the study.Choose ? subset of “locations” such thatthose outside the set have an ??? incident from ? member of the set, whichmeans that ?ll the “locations” can be “serviced’ by the membersof the set S. Let w(S) denote the sum of the weights of the members of S. Theproblem of finding such ? set S such that w(S) is minimized.

The relevant graphtheoretic concept is that of directed domination.3.6.  Conclusions   andOpen ProblemsDomination and other related topics in undirectedgraphs are extensively stud-ied,both theoretically and algorithmically. However, the corresponding topics ondigraphs have not received much attention, even though digraphs come up ??r?naturally in modelling real world problems. With this view in mind, we havemade an attempt to survey some of the existing results on domination relatedconcepts on digraphs.

We have also introduced some parameters on di-graphs analogous to dominationparameters on undirected graphs. As ? matter of fact, it seems that almost alldomination related problems on undirected graphs, if they make sense indigraphs, may be investigated. Algorithmic aspects of these problems ondigraphs will be another good area, of research.

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