Group-labeled Graph Research Articles

Packing cycles in undirected group-labelled graphs

We prove a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs (G,γ) where γ assigns to each edge of an undirected graph G an element of an abelian group Γ. As a consequence, we prove that Γ-nonzero cycles (cycles whose edge labels sum to a non-identity element of Γ) satisfy the half-integral Erdős-Pósa property, and we also recover a result of Wollan that if Γ has no element of order two, then Γ-nonzero cycles satisfy the Erdős-Pósa property. As another application, we prove that if m is an odd prime power, then cycles of length ℓmodm satisfy the Erdős-Pósa property for all integers ℓ. This partially answers a question of Dejter and Neumann-Lara from 1987 on characterizing all such integer pairs (ℓ,m).

Packing A-paths of length zero modulo a prime

It is known that A-paths of length 0 mod m satisfy the Erdős-Pósa property if m=2 or m=4, but not if m>4 is composite. We show that if p is prime, then A-paths of length 0 mod p satisfy the Erdős-Pósa property. More generally, in the framework of undirected group-labeled graphs, we characterize the abelian groups Γ and elements ℓ∈Γ for which the Erdős-Pósa property holds for A-paths of weight ℓ.

Finding a Shortest Non-Zero Path in Group-Labeled Graphs

We study a constrained shortest path problem in group-labeled graphs with nonnegative edge length, called the shortest non-zero path problem. Depending on the group in question, this problem includes two types of tractable variants in undirected graphs: one is the parity-constrained shortest path/cycle problem, and the other is computing a shortest noncontractible cycle in surface-embedded graphs. For the shortest non-zero path problem with respect to finite abelian groups, Kobayashi and Toyooka (2017) proposed a randomized, pseudopolynomial-time algorithm via permanent computation. For a slightly more general class of groups, Yamaguchi (2016) showed a reduction of the problem to the weighted linear matroid parity problem. In particular, some cases are solved in strongly polynomial time via the reduction with the aid of a deterministic, polynomial-time algorithm for the weighted linear matroid parity problem developed by Iwata and Kobayashi (2021), which generalizes a well-known fact that the parity-constrained shortest path problem is solved via weighted matching. In this paper, as the first general solution independent of the group, we present a rather simple, deterministic, and strongly polynomial-time algorithm for the shortest non-zero path problem. This result captures a common tractable feature behind the parity and topological constraints in the shortest path/cycle problem. The algorithm is based on Dijkstra’s algorithm for the unconstrained shortest path problem and Edmonds’ blossom shrinking technique in matching algorithms; this approach is inspired by Derigs’ faster algorithm (1985) for the parity-constrained shortest path problem via a reduction to weighted matching. Furthermore, we improve our algorithm so that it does not require explicit blossom shrinking, and make the computational time match Derigs’ one. In the speeding-up step, a dual linear programming formulation of the equivalent problem based on potential maximization for the unconstrained shortest path problem plays a key role.

A New Matroid Lift Construction and an Application to Group-Labeled Graphs

A well-known result of Brylawski constructs an elementary lift of a matroid $M$ from a linear class of circuits of $M$. We generalize this result by constructing a rank-$k$ lift of $M$ from a rank-$k$ matroid on the set of circuits of $M$. We conjecture that every lift of $M$ arises via this construction.
 We then apply this result to group-labeled graphs, generalizing a construction of Zaslavsky. Given a graph $G$ with edges labeled by a group, Zaslavsky's lift matroid $K$ is an elementary lift of the graphic matroid $M(G)$ that respects the group-labeling; specifically, the cycles of $G$ that are circuits of $K$ coincide with the cycles that are balanced with respect to the group-labeling. For $k \geqslant 2$, when does there exist a rank-$k$ lift of $M(G)$ that respects the group-labeling in this same sense? For abelian groups, we show that such a matroid exists if and only if the group is isomorphic to the additive group of a non-prime finite field.

Frame matroids, toric ideals, and a conjecture of White

Blasiak verified a conjecture of White for graphic matroids by showing that the toric ideal of a graphic matroid is generated by quadrics. In this paper, we extend this result to frame matroids satisfying a linearity condition. Such classes of frame matroids include graphic matroids, bicircular matroids, signed graphic matroids, and more generally frame matroids obtained from group-labelled graphs.

Finding a path with two labels forbidden in group-labeled graphs

The parity of the length of paths and cycles is a classical and well-studied topic in graph theory and theoretical computer science. The parity constraints can be extended to label constraints in a group-labeled graph, which is a directed graph with each arc labeled by an element of a group. Recently, paths and cycles in group-labeled graphs have been investigated, such as packing non-zero paths and cycles, where “non-zero” means that the identity element is a unique forbidden label. In this paper, we present a solution to finding an s – t path with two labels forbidden in a group-labeled graph. This also leads to an elementary solution to finding a zero s – t path in a Z 3 -labeled graph, which is the first nontrivial case of finding a zero path. This situation in fact generalizes the 2-disjoint paths problem in undirected graphs, which also motivates us to consider that setting. More precisely, we provide a polynomial-time algorithm for testing whether there are at most two possible labels of s – t paths in a group-labeled graph or not, and finding s – t paths attaining at least three distinct labels if exist. The algorithm is based on a necessary and sufficient condition for a group-labeled graph to have exactly two possible labels of s – t paths, which is the main technical contribution of this paper.

A Unified Erdős–Pósa Theorem for Constrained Cycles

A doubly group-labeled graph is an oriented graph with its edges labeled by elements of the direct sum of two groups $\Gamma_1,\Gamma_2$. A cycle in a doubly group-labeled graph is $(\Gamma_1,\Gamma_2)$-non-zero if it is non-zero in both coordinates. Our main result is a generalization of the Flat Wall Theorem of Robertson and Seymour to doubly group-labeled graphs. As an application, we determine all canonical obstructions to the Erd\H{o}s-P\'osa property for $(\Gamma_1,\Gamma_2)$-non-zero cycles in doubly group-labeled graphs. The obstructions imply that the half-integral Erd\H{o}s-P\'osa property always holds for $(\Gamma_1,\Gamma_2)$-non-zero cycles. Moreover, our approach gives a unified framework for proving packing results for constrained cycles in graphs. For example, as immediate corollaries we recover the Erd\H{o}s-P\'osa property for cycles and $S$-cycles and the half-integral Erd\H{o}s-P\'osa property for odd cycles and odd $S$-cycles. Furthermore, we recover Reed's Escher-wall Theorem. We also prove many new packing results as immediate corollaries. For example, we show that the half-integral Erd\H{o}s-P\'osa property holds for cycles not homologous to zero, odd cycles not homologous to zero, and $S$-cycles not homologous to zero. Moreover, the (full) Erd\H{o}s-P\'osa property holds for $S_1$-$S_2$-cycles and cycles not homologous to zero on an orientable surface. Finally, we also describe the canonical obstructions to the Erd\H{o}s-P\'osa property for cycles not homologous to zero and for odd $S$-cycles.

Count Matroids of Group-Labeled Graphs

A graph $G=(V,E)$ is called $(k,\ell)$-sparse if $|F|\leq k|V(F)|-\ell$ for any nonempty $F\subseteq E$, where $V(F)$ denotes the set of vertices incident to $F$. It is known that the family of the edge sets of $(k,\ell)$-sparse subgraphs forms the family of independent sets of a matroid, called the $(k,\ell)$-count matroid of $G$. In this paper we shall investigate lifts of the $(k,\ell)$-count matroid by using group labelings on the edge set. By introducing a new notion called near-balancedness, we shall identify a new class of matroids, where the independence condition is described as a count condition of the form $|F|\leq k|V(F)|-\ell +\alpha_{\psi}(F)$ for some function $\alpha_{\psi}$ determined by a given group labeling $\psi$ on $E$.

Finding a Shortest Non-zero Path in Group-Labeled Graphs via Permanent Computation

A group-labeled graph is a directed graph with each arc labeled by a group element, and the label of a path is defined as the sum of the labels of the traversed arcs. In this paper, we propose a polynomial time randomized algorithm for the problem of finding a shortest s-t path with a non-zero label in a given group-labeled graph (which we call the Shortest Non-Zero Path Problem). This problem generalizes the problem of finding a shortest path with an odd number of edges, which is known to be solvable in polynomial time by using matching algorithms. Our algorithm for the Shortest Non-Zero Path Problem is based on the ideas of Bj?rklund and Husfeldt (Proceedings of the 41st international colloquium on automata, languages and programming, part I. LNCS 8572, pp 211---222, 2014). We reduce the problem to the computation of the permanent of a polynomial matrix modulo two. Furthermore, by devising an algorithm for computing the permanent of a polynomial matrix modulo $$2^r$$2r for any fixed integer r, we extend our result to the problem of packing internally-disjoint s-t paths.

Packing $A$-Paths in Group-Labelled Graphs via Linear Matroid Parity

Mader's disjoint ${\cal S}$-paths problem is a common generalization of non-bipartite matching and Menger's disjoint paths problems. Lovasz [J. Combin. Theory Ser. B, 28 (1980), pp. 208--236]) proposed a polynomial-time algorithm for this problem through a reduction to matroid matching. A more direct reduction to the linear matroid parity problem was given later by Schrijver [Combinatorial Optimization: Polyhedra and Efficiency, Springer-Verlag, Berlin, 2003], which led to faster algorithms. As a generalization of Mader's problem, Chudnovsky et al. [Combinatorica, 26 (2006), pp. 521--532] introduced a framework of packing non-zero $A$-paths in group-labelled graphs and proved a min-max theorem. Chudnovsky, Cunningham, and Geelen [Combinatorica, 28 (2008), pp. 145--161] provided an efficient combinatorial algorithm for this generalized problem. On the other hand, Pap [Combinatorica, 27 (2007), pp. 247--251] introduced a framework of packing non-returning $A$-paths as a further generalization. In this paper...

Symmetry-forced rigidity of frameworks on surfaces

A fundamental theorem of Laman characterises when a bar-joint framework realised generically in the Euclidean plane admits a non-trivial continuous deformation of its vertices. This has recently been extended in two ways. Firstly to frameworks that are symmetric with respect to some point group but are otherwise generic, and secondly to frameworks in Euclidean 3-space that are constrained to lie on 2-dimensional algebraic varieties. We combine these two settings and consider the rigidity of symmetric frameworks realised on such surfaces. First we establish necessary conditions for a framework to be symmetry-forced rigid for any group and any surface by setting up a symmetry-adapted rigidity matrix for such frameworks and by extending the methods in Jordan et al. (2012) to this new context. This gives rise to several new symmetry-adapted rigidity matroids on group-labelled quotient graphs. In the cases when the surface is a sphere, a cylinder or a cone we then also provide combinatorial characterisations of generic symmetry-forced rigid frameworks for a number of symmetry groups, including rotation, reflection, inversion and dihedral symmetry. The proofs of these results are based on some new Henneberg-type inductive constructions on the group-labelled quotient graphs that correspond to the bases of the matroids in question. For the remaining symmetry groups in 3-space—as well as for other types of surfaces—we provide some observations and conjectures.

Balanced group-labeled graphs

A group-labeled graph is a graph whose vertices and edges have been assigned labels from some abelian group. The weight of a subgraph of a group-labeled graph is the sum of the labels of the vertices and edges in the subgraph. A group-labeled graph is said to be balanced if the weight of every cycle in the graph is zero. We give a characterization of balanced group-labeled graphs that generalizes the known characterizations of balanced signed graphs and consistent marked graphs. We count the number of distinct balanced labellings of a graph by a finite abelian group Γ and show that this number depends only on the order of Γ and not its structure. We show that all balanced labellings of a graph can be obtained from the all-zero labeling using simple operations. Finally, we give a new constructive characterization of consistent marked graphs and markable graphs, that is, graphs that have a consistent marking with at least one negative vertex.

FPT algorithms for path-transversal and cycle-transversal problems

We study the parameterized complexity of several vertex- and edge-deletion problems on graphs, parameterized by the number p of deletions. The first kind of problems are separation problems on undirected graphs, where we aim at separating distinguished vertices in a graph. The second kind of problems are feedback set problems on group-labelled graphs, where we aim at breaking nonnull cycles in a graph having its arcs labelled by elements of a group. We obtain new FPT algorithms for these different problems, relying on a generic O ∗ ( 4 p ) algorithm for breaking paths of a homogeneous path system.

Excluding a group-labelled graph

This paper contains a first step towards extending the Graph Minors Project of Robertson and Seymour to group-labelled graphs. For a finite abelian group Γ and Γ-labelled graph G, we describe the class of Γ-labelled graphs that do not contain a minor isomorphic to G.

An algorithm for packing non-zero A-paths in group-labelled graphs

Let G = (V, E) be an oriented graph whose edges are labelled by the elements of a group Γ and let A ? V. An A-path is a path whose ends are both in A. The weight of a path P in G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in P. (If Γ is not abelian, we sum the labels in their order along the path.) We give an efficient algorithm for finding a maximum collection of vertex-disjoint A-paths each of non-zero weight. When A = V this problem is equivalent to the maximum matching problem.

Packing Non-Returning A-Paths*

Chudnovsky et al. gave a min-max formula for the maximum number of node-disjoint nonzero A-paths in group-labeled graphs [1], which is a generalization of Mader's theorem on node-disjoint A-paths [3]. Here we present a further generalization with a shorter proof. The main feature of Theorem 2.1 is that parity is “hidden” inside $$\ifmmode\expandafter\hat\else\expandafter\^\fi{v}$$, which is given by an oracle for non-bipartite matching.