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Polynomials for Directed Graphs Polynomials for Directed Graphs Gary Gordon and Lorenzo Traldi Departmen t of Mathematics Lafayette College Easton, PA 18042 Abstract Several polynomials are defined on directed graphs and rooted directed graphs which are all analogous to the Tutte polynomial of an undirected graph. Various interrelations between these polynomials are explored. Many of them have the number of maximal di rected trees (arbores cences) as an evaluation, and are related to the one- variable greedoid polynomial ~G(Y). Applications to reliability are also examined. 1. Introduction. The central theme of this study is the attempt to generalize the Tutte polynomial of an undirected graph to a polynomial invariant of a directed graph, especially one having a distinguished vertex (a rooted digraph ). It does not seem that there is any one right" such generalization; rather, different definitions of the Tutte polynomial suggest different directed polynomials. One definition of the Tutte polynomial of an undirected graph G depends on a rank function on subsets of E ( G), the edge-set of G. For A ~ E (G) the (circuit ) rank of A, r (A), is the cardinality of the largest acyclic subset of A_ The corank-nullity polynomial f( Gj t, z) (also called the Whitney polynomiaf) is defin ed by (T1) f(Gjt,z ) == L tr(E(G» - r(A) z IAI-r(A). A~E (G) In Section 2, we !lese three differen t rank functions to define polynomials on rooted digraphs via (T1). (By a rank function" we simply mean any function mapping sets of edges of directed graphs to non-negative integers. ) One of these three is the rank fun ction of the directed branchi ng greedoid (1] associated to a rooled digraph D, while the second is the basis rank fu nction of the same greedoid. Thus , these first two rank functions yield two-variable polynomial invariants for arbitrary greedoids. The third rank fu nction considered is not associated with any greedoid, though it is still associated with the notion of reachability of vertices of D fr om its root. All of these polynomials generalize the one-variable greedoid polynomial of [1] when the greedoid is the di rected branching greedoid of a rooted digraph. CONGRESSUS NUMERANTIUM 94(1993), pp.187-201 The definition of the corank-nullity polynomial of an undirected graph can be rephrased without using a rank function. For this formulation, define a subgraph A of G to be vertex-spanning if it contains the entire vertex-set V(C) of C, and let 5 denote the collection of all such subgraphs; we will often use the same letter A to refer to both a vertex-spanning subgraph and its edge-set. Then if c(A ) denotes the number of connected components of A, (T2) f( c; t, z) == E tc(A)-c(G) zIAI+c(A)-IV(G)I. A C:;; E (G) Since r(A) W(C)I - etA), (T2) is a direct translation of (Tl). We mention it here because it is natural to investigate the notions of "number of components" associated to various rank functions. For instance, the notion associated to the rank function of the directed branching greedoid is that the number of components of a rooted digraph is one more than the number of vertices not reachable from the root, wh il e the notion associated to the third rank function considered in Section 2 is that the number of components of a rooted digraph is the minimum number of edges that can be inserted into the digraph so that all its vertices will be reachable from its root. Also, it often h ppens that a certain notion of "number of components" can be more natural than the associated rank function. For instance, in Section 3 we menti n two polynomials t hat are associated wi th strong connectedness rather than reachability, and for these polynomials there are no "natural" descriptions of the associa ted rank func tions. The recursive definition of t he Tutte polynomial is very useful, since it allows for natural inductive proofs. If e is an edge of C then the deletion C - e is simply the graph C wi th e erased (deleted). The cont raction G / e is the graph obtained from C - e by identifyin g the endpoints of e. The Tutte polynomial can be defined by (T3) (a) ftC) = 1 ifC has no edges, (b ) I ( G) = f ( C - e) + f( G / e) if e is ne ither an isthmus nor a loop, (c) ft C ) = xf(C /e) if e is an isthmu,s, and (d) f tC) = yf(C - e) ife is a loop. Note that (T3) (b), (c), and (d) can be replaced by the following single condition. (e) For any e, ftC ) = (y - l )IV(G /e)I+1-I V (G )lf(C/e) + (x - ly(G)-r(G-e) f (C - e). (T 3) is equivalent to (Tl) and (T2) under the substitution x = t +1, II = z + 1. It seems di ffi cult to defi ne a "directed isthmus" that can be u.sed in (T 3) to recursively define a polynomial that is independent of the order in which edges are deleted and contracted. In Section 4 we give two exa.mples of recursively defined polynomials whi ch depend on a. given ordering of t he edges or vertices. One can also define the ordinary Tutte polynomial in terms of activi ties (see [2] and [9]) or generalized activities (see [7]) . Although several of the polynomials we defi ne for rooted digraphs can be described in term.s of analogous activities, these 188 descriptions are complicated and do not seem to contribute much to understanding the polynomials. In Section 5 we extend the polynomials to weighted digraphs; several can then be evaluated to yield the determinant of the Kirch hoff matri x. We also discuss the application of these pol ynomials to some reliability problems . T he polynomials yield no t only information about the likelihood that a directed network with imperfect edges will remain fully operational, but also illformation about the expected number of edges that will fai l, the expected number of repairs that will have to be made to restore a ne twork to operational status, and the expected cos t of making those repaIrs. Before considering this Introduction cnmplete we should take the opportunity to thank Lafayette College for its support : during part of the time this research was being done Gary Gordon was supported by a research fellowship, and Lorenzo Traldi was on a sabbatical leave. 2. Polynomials for rooted digraphs. In this section, we defi ne three polynomials for rooted digraphs via (Tl). We fi rst define the branching greedoid independence rank rl (A ) and basis rank r~ ( A ) , and also the rooted forest rank r3( A ), of a subset A of the edge-set of a rooted digraph (D, *). (Here .. represen ts t he root vertex. ) A subgrapb of D is a '" rooted arborescence if for every vertex v of T, there is a uni que directed path in T from * to v. T hus T is a di rected tree, rooted at • . (See Figure la.) A spanning. rooted arborescfnCf is a • rooted arborescence which is incident with every vertex of D. In general, a digraph need not have any spanning . rooted arborescence; even if it doesn't , it is still true tbat all maximal * rooted arborescences have the same number of edges. A • rooted forest of arborescences is a vertex-disjoi nt union of arborescences rooted at ., VI, V2t ••• for some vertices VI, V2, .... (In Figure 1b, the heavy lines give a * rooted forest of arborescenc s. ) We will often fail to distinguish be ween a • rooted forest of arborescences and its edge-set. We now defi ne rank fun ctions r l, r2, and r3 by (2.1) rl(A) == max{IT II T ~ A is a * rooted arbo rescence}, (2.2) r2(A) == 'Dax {IT nAil T ~ E(D) is a * roo ted arborescence}, and (2.3) r3 (A) == max{IFI I F ~ A is a. root ed forest of arborescences }. Proposition 2.1. For every A ~ E(D), rt{A) ~ r2 (A) ~ r3 (A); also, rdD) rd D). Proof. Since any • rooted arboresc nce can be extended to a maximal . rooted arborescence, rl (A ) ~ r2( A). Furthermore, since any subset of a maximal * rooted arborescence is a • rooted forest of arborescences, r2 (A ) :5 r3 (A). 189 .... The above inequali ties are not generally reversible. For instance, in Figure 2a if A = {a, c, e} then rl( A) = 1, r2( A ) = 2, and r3( A) = 3. It is a routine veri fi cation to demonstrate that rl is a greedoid rank function. In general , it is not a matroid rank fun ction. For examp le, let A = { b, c d} and x = a in the digraph of Figure 2a. Then rdA) = 0, but rl (A U {x}) = 4. It is also easy to verify that each of r2 and rJ satis fi es the defini tion of a matroid rank function except for semimodulari ty. For example, in Figure 2b for i = 2 or 3 we have r;( {c}) = r;( {c,d}) = r.( {a, c}) = 1, but r;({a,c,d}) = 2. Imitating defi nitions of matroid theory (see [10J for an account), for 1 Si S 3 we define A ~ E( D ) to he i-spanning if ri tA) = r.(E (D)), i-independent if r;(A ) = IAI. and an i-basis if it is both i-spanning and i-independent. The following proposition is an immediate consequence of Definitions (2.1), (2.2), and (2 .3 ).
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