View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 310 (2010) 2380–2388 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc A branching greedoid for multiply-rooted graphs and digraphs Lauren Eaton a, Steven J. Tedford b,∗ a Department of Mathematics, Franklin & Marshall College, Lancaster, PA, United States b Department of Mathematics and Computer Science, Misericordia University, Dallas, PA, United States article info a b s t r a c t Article history: Two combinatorial structures which describe the branchings in a graph are graphic ma- Received 22 May 2009 troids and undirected branching greedoids. We introduce a new class of greedoids which Received in revised form 14 April 2010 connects these two structures. We also apply these greedoids to directed graphs to con- Accepted 11 May 2010 sider a matroid defined on a directed graph. Finally, we obtain a formula for the greedoid Available online 1 June 2010 characteristic polynomial for multiply-rooted directed trees which can be determined from the vertices. Keywords: ' 2010 Elsevier B.V. All rights reserved. Greedoid Multiply-rooted directed graph Characteristic polynomial 1. Introduction When considering problems in graph theory, it is not unusual to look at the branching structure in the graph. In fact, there are two well-known combinatorial structures which describe this branching structure—graphic matroids and undirected branching greedoids. While both of these structures consider the branchings in a graph, their underlying viewpoints are extreme opposites. In a graphic matroid, any subset of the edges (as long as it is acyclic) is considered feasible, while in an undirected branching greedoid, only connected acyclic subsets which contain a particular root vertex are considered feasible. These two structures are the two extreme members of a class of larger combinatorial structures which we call multiply-rooted undirected branching greedoids. By considering rooted arborescences instead of rooted trees, the directed branching greedoid can be defined on a rooted directed graph. These greedoids lie inside the larger class of multiply-rooted directed branching greedoids which we introduce. We will discuss the other extreme member of this class, which will be a matroid defined on the directed graph. Finally, we restrict ourselves to multiply-rooted directed trees and show that the greedoid characteristic polynomial can be determined combinatorially. 2. Definitions We start with the necessary graph theoretical and greedoid theoretical terminologies. We will use TnU to denote the set f g V ! ⊆ D P 1; 2;:::; n for any nonnegative integer n. Also, as is common, if f E R, then for any S E, we define f .S/ e2S f .e/. 2.1. Graph theory A graph Γ consists of a set V .Γ / of vertices and a set E.Γ / of edges, where each edge e 2 E.Γ / is associated to a pair of (not necessarily distinct) vertices v1, v2. In a directed graph Λ, the edges are associated to an ordered pair of vertices ∗ Corresponding author. E-mail address: [email protected] (S.J. Tedford). 0012-365X/$ – see front matter ' 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2010.05.007 L. Eaton, S.J. Tedford / Discrete Mathematics 310 (2010) 2380–2388 2381 (a) F1. (b) F2. (c) F3. Fig. 1. One feasible and two nonfeasible sets in multiply-rooted directed graphs. .v1; v2/ and a directed edge e is said to be directed from v1 to v2. We will say that Γ (Λ) is multiply rooted—with root-set R—if R ⊆ V .Γ / (R ⊆ V .Λ/) (if R D fvg, then the (di)graph is said to be rooted at v). If no ambiguity will occur, we will use V and E to denote the corresponding sets. We will use TΓ UR to denote a graph Γ with root-set R and TΛUR to denote a directed graph Λ with root-set R. A path in Γ is a sequence .v0; e1; : : : ; vn−1; en; vn/ with each edge ei incident to the vertices vi−1 and vi such that no vertex is repeated. A subgraph of Γ is acyclic if every path between two vertices in the subgraph is unique. A subgraph of Γ is a tree if it is acyclic and connected. For directed graphs, a path is directed if every edge ei is directed from vi−1 to vi. An arborescence F in TΛUR is a tree in TΛUR containing a root r 2 R with the property that, for every vertex v 2 V .F/ with v 6D r, the path in F from r to v is directed. See [1] for further graph theoretical definitions. 2.2. Greedoid theory A finite set E and a non-empty collection of subsets F form a greedoid G D .E; F / if: 1. For each non-empty F 2 F , there exists an e 2 F such that F n e 2 F . 2. If F1; F2 2 F and jF1j > jF2j then there exists an e 2 F1 n F2 with F2 [ e 2 F . The sets F 2 F .G/ are called the feasible sets of G. If e 2 E is not in any feasible subset, then e is called a g-loop of G. Given S ⊆ E, the rank of S, denoted rG.S/, is defined by: rG.S/ D maxfjFj V F 2 F I F ⊆ Sg: For ease of notation, we will let corG.S/ D rG.E/−rG.S/ and nulG.S/ D jSj−rG.S/. If there is no confusion about the greedoid under discussion, we will use r.S/, cor.S/, and nul.S/ to denote the rank, corank, and nullity functions respectively. Next, we define the operations of deletion and contraction for greedoids. Given a set S ⊆ E, we define the deletion of S as the greedoid G n S D .E n S; F n S/ where F n S D fF ⊆ E n S V F 2 F g. Given a feasible set F, we define the contraction of F as the greedoid G=F D .E n F; F =F/, where F =F D fH ⊆ E n F V H [ F 2 F g. If we are deleting or contracting a single element e, we will write G n e and G=e to denote G n feg and G=feg respectively. Given two greedoids G1 D .E1; F1/ and G2 D .E2; F2/ with E1 \ E2 D;, the direct sum of G1 and G2 is the greedoid G1 ⊕ G2 D .E1 [ E2; F1 t F2/ with feasible sets defined by F1 t F2 D fF1 [ F2 V F1 2 F1 and F2 2 F2g: See [4] for further greedoid theoretical definitions. 3. Multiply-rooted branching greedoids Given a multiply-rooted graph TΓ UR, we define the multiply-rooted undirected branching greedoid, G.TΓ UR/ on E D E.Γ / by defining F ⊆ E to be feasible if the graph induced by F is acyclic with the property that each component of Γ TFU contains at least one root. If R D frg, then G.TΓ UR/ is called the undirected branching greedoid of Γ . Given a multiply-rooted directed graph TΛUR, we define the multiply-rooted directed branching greedoid, G.TΛUR/ on E D E.Λ/ by defining F ⊆ E to be feasible if the graph induced by F is acyclic and there exists a partition of F D F1 [ F2 [···[ Fn such that the graph induced by Fi is an arborescence rooted at a vertex in R. In Fig. 1, F1 is feasible, but both F2 and F3 are not (vertices in R are denoted by a •). If R D frg, then G.TΛUR/ is called the directed branching greedoid of Λ. Lemma 1. Given a finite multiply-rooted graph TΓ UR or a finite multiply-rooted digraph TΛUR,G.TΓ UR/ (G.TΛUR/) is a greedoid. The proof of this lemma mirrors the proof for undirected and directed branching greedoids. In an undirected graph, if R D fvg for some vertex v, then G.TΓ UR/ is, by definition, the undirected branching greedoid for Γ rooted at v. Similarly, in a directed graph Λ, if R D fvg, then G.TΛUR/ is, by definition, the directed branching greedoid for Λ rooted at v. On the other hand, if R D V , then G.TΓ UV / must be the graphic matroid of Γ . Similarly, G.TΛUV / is also a matroid. It is clear that making every vertex of a directed graph a root is essentially equivalent to removing the directions from all the edges of the digraph. This supports the belief that there is no natural matroid describing the branching structure in a directed graph which also includes the directions on the edges. For the sequel, we will discuss properties for multiply-rooted directed graphs. Although many results may also apply to undirected graphs, we will not consider that here. 2382 L. Eaton, S.J. Tedford / Discrete Mathematics 310 (2010) 2380–2388 (a) Λ. (b) ΛΠ .v/. (c) ΛN .v/. (d) ΛC.v/. Fig. 2. Different vertex splittings. As is to be expected, structure in a multiply-rooted digraph is mirrored by properties in its multiply-rooted branching greedoid. In particular, if the digraph has a cut-vertex, then that vertex can be split and it will not change the greedoid in many circumstances. In particular, the greedoid never changes when the cut-vertex is either a root or a sink. We look at these and similar properties here. The first two results are easy to show.