Maximal Pivots on Graphs with an Application to Gene Assembly

📝 Abstract

We consider principal pivot transform (pivot) on graphs. We define a natural variant of this operation, called dual pivot, and show that both the kernel and the set of maximally applicable pivots of a graph are invariant under this operation. The result is motivated by and applicable to the theory of gene assembly in ciliates.

💡 Analysis

We consider principal pivot transform (pivot) on graphs. We define a natural variant of this operation, called dual pivot, and show that both the kernel and the set of maximally applicable pivots of a graph are invariant under this operation. The result is motivated by and applicable to the theory of gene assembly in ciliates.

📄 Content

arXiv:0909.3789v3 [math.CO] 14 Oct 2010 Maximal Pivots on Graphs with an Application to Gene Assembly Robert Brijder∗, Hendrik Jan Hoogeboom Leiden Institute of Advanced Computer Science, Leiden University, The Netherlands Abstract We consider principal pivot transform (pivot) on graphs. We define a natural variant of this operation, called dual pivot, and show that both the kernel and the set of maximally applicable pivots of a graph are invariant under this oper- ation. The result is motivated by and applicable to the theory of gene assembly in ciliates. Keywords: principal pivot transform, algebraic graph theory, overlap graph, gene assembly in ciliates

1. Introduction The pivot operation, due to Tucker [18], partially (component-wise) inverts a given matrix. It appears naturally in many areas including mathematical programming and numerical analysis, see [17] for a survey. Over F2 (which is the natural setting to consider for graphs), the pivot operation has, in addition to matrix and graph interpretations [11], also an interpretation in terms of delta matroids [1]. In this paper we define the dual pivot, which has an identical effect on graphs as the (regular) pivot, however the condition for it to be applicable differs. The main result of the paper is that any two graphs in the same orbit under dual pivot have the same family of maximal pivots (cf. Theorem 16), i.e., the same family of maximally partial inverses of that matrix. This result is obtained by combining each of the aforementioned interpretations of pivot. This research is motivated by the theory of gene assembly in ciliates [9], which is recalled in Section 7. Without the context of gene assembly this main result (Theorem 16) is surprising; it is not found in the extensive literature on pivots. It fits however with the intuition and results from the string based model of gene assembly [4], and in this paper we formulate it for the more ∗Corresponding author Email address: rbrijder@liacs.nl (Robert Brijder) Preprint submitted to Elsevier July 10, 2021 general graph based model. It is understood and proven here using completely different techniques, algebraical rather than combinatorial.
2. Notation and Terminology The field with two elements is denoted by F2. Our matrix computations will be over F2. Hence addition is equal to the logical exclusive-or, also denoted by ⊕, and multiplication is equal to the logical conjunction, also denoted by ∧. These operations carry over to sets, e.g., for sets A, B ⊆V and x ∈V , x ∈A ⊕B iff(x ∈A) ⊕(x ∈B). A set system is a tuple M = (V, D), where V is a finite set and D ⊆2V is a set of subsets of V . Let min(D) (max(D), resp.) be the family of minimal (maximal, resp.) sets in D w.r.t. set inclusion, and let min(M) = (V, min(D)) (max(M) = (V, max(D)), resp.) be the corresponding set systems. Let V be a finite set, and A be a V × V -matrix (over an arbitrary field), i.e., A is a matrix where the rows and columns of A are identified by elements of V . Therefore, e.g., the following matrices with V = {p, q} are equal:  p q p 1 1 q 0 1  and  q p q 1 0 p 1 1  . For X ⊆V , the principal submatrix of A w.r.t. X is denoted by A[X], i.e., A[X] is the X × X-matrix obtained from A by restricting to rows and columns in X. Similarly, we define A\X = A[V \X]. Notions such as matrix inversion A−1 and determinant det(A) are well defined for V × V -matrices. By convention, det(A[∅]) = 1. A set X ⊆V is called dependent in A iffthe columns of A corresponding to X are linearly dependent. We define PA = (I, D) to be the partition of 2V such that D (I, respectively) contains the dependent (independent, respectively) subsets of V in A. By convention, ∅∈I. The sets in max(I) are called the bases of A. We have that PA = (I, D) is uniquely determined by max(I) (and the set V ). Similarly, PA is uniquely determined by min(D) (and the set V ). These properties are specifically used in matroid theory, where a matroid may be described by its independent sets (V, I), by its family of bases (V, max(I)), or by its circuits (V, min(D)). Moreover, for each basis X ∈max(I), |X| is equal to the rank r of A. We consider undirected graphs without parallel edges, however we do allow loops. For a graph G = (V, E) we use V (G) and E(G) to denote its set of vertices V and set of edges E, respectively, where for x ∈V , {x} ∈E iffx has a loop. For X ⊆V , we denote the subgraph of G induced by X as G[X]. With a graph G one associates its adjacency matrix A(G), which is a V ×V - matrix (au,v) over F2 with au,v = 1 iff{u, v} ∈E. The matrices corresponding to graphs are precisely the symmetric F2-matrices; loops corresponding to diagonal 1’s. Note that for X ⊆V , A(G[X]) = (A(G))[X]. 2 Over F2, vectors indexed by V can be identified with subsets of V , and a V × V -matrix defines a linear transformation on subsets of V . The kernel (also called null space) of a matrix A, denoted by ker(A) is determined by those linear combinations of column vectors of A that sum up to th

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