Biased Graph Research Articles

Bias matroids with unique graphical representations

Given a 3-connected biased graph Ω with three node-disjoint unbalanced circles, at most one of which is a loop, we describe how the bias matroid of Ω is uniquely represented by Ω .

Quasigroup associativity and biased expansion graphs

We present new criteria for a multary (or polyadic) quasigroup to be isotopic to an iterated group operation. The criteria are consequences of a structural analysis of biased expansion graphs. We mention applications to transversal designs and generalized Dowling geometries.

The Pair Completion algorithm for the Homogeneous Set Sandwich Problem

A homogeneous set is a non-trivial module of a graph, i.e., a non-empty, non-unitary, proper vertex subset such that all its elements present the same outer neighborhood. Given two graphs G 1 ( V , E 1 ) and G 2 ( V , E 2 ) , the Homogeneous Set Sandwich Problem (HSSP) asks whether there exists a graph G S ( V , E S ) , E 1 ⊆ E S ⊆ E 2 , which has a homogeneous set. This paper presents an algorithm that uses the concept of bias graph [S. Tang, F. Yeh, Y. Wang, An efficient algorithm for solving the homogeneous set sandwich problem, Inform. Process. Lett. 77 (2001) 17–22] to solve the problem in O ( n min { | E 1 | , | E ¯ 2 | } log n ) time, thus outperforming the other known HSSP deterministic algorithms for inputs where max { | E 1 | , | E ¯ 2 | } = Ω ( n log n ) .

An efficient algorithm for solving the homogeneous set sandwich problem

A set H of vertices of graph G(V,E) is a homogeneous set if each vertex in V\\H is either adjacent to all vertices of H or to none of the vertices in H, where V and E are the vertex set and edge set, respectively, of graph G. A graph G s(V,E s) is called a sandwich graph for the pair of graphs G(V,E) and G t(V,E t) if E t⫅E s⫅E . The homogeneous set sandwich problem is to determine whether there exists a sandwich graph for the pair of graphs G and G t such that there is a homogeneous set in G s . In this paper, we shall propose an O(Δn 2) time algorithm for solving the homogeneous set sandwich problem, where Δ is the maximum degree of G. Furthermore, a surprising result comes from the bias graph which is an auxiliary graph for reducing the homogeneous set sandwich problem to the problem of finding strongly connected components. Using the bias graph, we can easily find all homogeneous sets for a sandwich graph problem with the same time complexity, i.e., O(Δn 2) .

Supersolvable Frame-matroid and Graphic-lift Lattices

A geometric lattice is a frame if its matroid, possibly after enlargement, has a basis such that every atom lies under a join of at most two basis elements. Examples include all subsets of a classical root system. Using the fact that finitary frame matroids are the bias matroids of biased graphs, we characterize modular coatoms in frames of finite rank and we describe explicitly the frames that are supersolvable. We apply the characterizations to three kinds of example. A geometric lattice is a graphic lift if it can be extended to contain an atom whose upper interval is graphic. We characterize modular coatoms in and supersolvability of graphic lifts of finite rank and we examine families analogous to the frame examples.

Biased Graphs .III. Chromatic and Dichromatic Invariants

A biased graph Ω consists of a graph Γ and a class of circles in Γ (edge sets of simple, closed paths), called balanced, such that no theta subgraph contains exactly two balanced circles. An edge set is balanced if (simplifying slightly) every circle in it is balanced. Biased graphs generalize ordinary graphs, which behave like biased graphs in which every circle is balanced. We define and study the chromatic, dichromatic, and Whitney number polynomials of a biased graph, which generalize those of an ordinary graph. We employ an algebraic definition since not all biased graphs can be colored. We show that the polynomials enjoy many properties that are familiar in ordinary graph theory, such as convolutional and partition expansions, close connections with the bias and lift matroids of Ω, and deletion-contraction invariance of the dichromatic polynomial. They also have the novel feature of being reducible to more readily computable related polynomials that have no analogs in ordinary graph theory. We apply our results to evaluate Whitney numbers and other invariants of the bias and lift matroids, to characterize the biased graphs which have an unbalanced edge at every node and whose bias matroid is a series-parallel network, and to calculate the invariants of some types of biased graphs, such as those where no circle is balanced and some which are similar to Dowling lattices and classical root systems. For the latter we also characterize supersolvability, a matroid property which implies the characteristic polynomial has positive integral roots.

Frame Matroids and Biased Graphs

A frame matroid is any submatroid of a matroid in which each point belongs to a line spanned by a fixed basis. A biased graph is a graph with certain polygons called balanced, no theta graph containing exactly two balanced polygons. We prove that certain matroids, called bias matroids, of biased graphs are identical to the finitary frame matroids. As an application we deduce two simple characterizations of frame matroids and some facts about planar forbidden minors for bias matroids.

Deterministic Graph Games and a Probabilistic Intuition

There is a close relationship between biased graph games and random graph processes. In this paper, we develop the analogy and give further interesting instances.

Characterizing bias matroids

del Greco, J.G., Characterizing bias matroids, Discrete Mathematics 103 (1992) 153–159. In this paper, a set of necessary and sufficient conditions for a matroid to be the bias matroid of a given biased graph is obtained. These conditions are reminiscent of a 1981 result of Seymour that characterized when a matroid is the polygon matroid of a graph.

Biased graphs. II. The three matroids

A biased graph Ω consists of a graph Γ and a class B of circles (simple, closed paths) in Γ, called balanced circles, such that no theta subgraph contains exactly two balanced circles. The bias matroid G(Ω) is a finitary matroid on the edge set E of Ω whose circuits are the balanced circles and the minimal connected edge sets of cyclomatic number two containing no balanced circle. We prove that these circuits define a matroid and we establish cryptomorphic definitions and other properties. Another finitary matroid on E, the lift matroid L(Ω), and its one-point extension the complete lift matroid L 0(Ω) , are obtained from the abstract matroid lift construction applied to the graphic matroid G( Γ) and the class B . The circuits of L(Ω) are the balanced circles and the minimal edge sets of cyclomatic number two (not necessarily connected) containing no balanced circle. We develop cryptomorphisms and other properties of L 0(Ω) and L(Ω). There is no completely general construction rule, besides the bias and lift constructions, which assigns to each biased graph a matroid intermediate (in the sense of independent sets) between G and L and which respects subgraphs. G(Ω) has an infinitary analog for infinite graphs generalizing Klee's infinitary bicircular matroid and the Bean-Higgs infinitary graphic matroid. Whether L(Ω) has an infinitary analog is unclear.

Biased graphs whose matroids are special binary matroids

Abiased graph is a graph together with a class of polygons such that no theta subgraph contains exactly two members of the class. To a biased graphΩ are naturally associated three edge matroids:G(Ω), L(Ω), L0(Ω). We determine all biased graphs for which any of these matroids is isomorphic to the Fano plane, the polygon matroid ofK4,K5 orK3,3, any of their duals, Bixby's regular matroidR10, or the polygon matroid ofKm form > 5. In each case the bias is derived from edge signs. We conclude by finding the biased graphsΩ for whichL0(Ω) is not a graphic [or, regular matroid but every proper contraction is.

Biased graphs. I. Bias, balance, and gains

A biased graph is a graph together with a class of cirles (simple closed paths), called balanced, such that no theta subgraph contains exactly two balanced circles. A gain graph is a graph in which each edge has a gain (a label from a group so that reversing the direction inverts the gain); a circle is balanced if its edge gain product is 1; this defines a biased graph. We initiate a series devoted to biased graphs and their matroids. Here we study properties of balance and also subgraphs and contractions of biased and gain graphs.

A generalisation of the matroid lift construction

This paper introduces a general matroid-theoretic construction which includes, as special cases, elementary lifts of matroids and bias matroids of biased graphs. To perform the construction on a matroid M M , it is necessary (but not sufficient) to have a submodular function inducing M M . Elementary lifts are obtained when the submodular function chosen is the rank function of M M . We define what is meant by a k k -induced matroid. These matroids simultaneously generalise matroids of graphs, transversal matroids and Dilworth truncations. They are induced by a particularly natural class of submodular functions. The effect of the above construction on k k -induced matroids using these natural submodular functions is studied. Results on minors of k k -induced matroids and the matroids obtained from them using the construction are given.

Vertices of Localized Imbalance in a Biased Graph

A biased graph consists of a graph $\Gamma$ and a subclass $\mathcal {B}$ of the polygons of $\Gamma$, such that no theta subgraph of $\Gamma$ contains exactly two members of $\mathcal {B}$. A subgraph is balanced when all its polygons belong to $\mathcal {B}$. A vertex is a balancing vertex if deleting it leaves a balanced graph. We give a construction for unbalanced biased graphs having a balancing vertex and we show that an unbalanced biased graph having more than one balancing vertex is an unbalanced series or parallel connection of balanced graphs.

The biased graphs whose matroids are binary

A biased graph Φ consists of a graph and a class of distinguished polygons such that no theta subgraph contains exactly two distinguished polygons. There are three matroids naturally associated with Φ: the bias matroid G( Φ), the lift matroid L( Φ), and the complete lift L 0( Φ). We characterize those Φ for which any of these matroids is binary.

Vertices of localized imbalance in a biased graph

A biased graph consists of a graph Γ \Gamma and a subclass B \mathcal {B} of the polygons of Γ \Gamma , such that no theta subgraph of Γ \Gamma contains exactly two members of B \mathcal {B} . A subgraph is balanced when all its polygons belong to B \mathcal {B} . A vertex is a balancing vertex if deleting it leaves a balanced graph. We give a construction for unbalanced biased graphs having a balancing vertex and we show that an unbalanced biased graph having more than one balancing vertex is an unbalanced series or parallel connection of balanced graphs.