Non-separating Cycles Research Articles

Approximating Maximum Integral Multiflows on Bounded Genus Graphs

We devise the first constant-factor approximation algorithm for finding an integral multi-commodity flow of maximum total value for instances where the supply graph together with the demand edges can be embedded on an orientable surface of bounded genus. This extends recent results for planar instances. Our techniques include an uncrossing algorithm, which is significantly more difficult than in the planar case, a partition of the cycles in the support of an LP solution into free homotopy classes, and a new rounding procedure for freely homotopic non-separating cycles.

Contractible Edges and Peripheral Cycles in 3-Connected Graphs

Peripheral cycles (induced non-separating cycles) in a general 3-connected graph are analogous to the faces of a polyhedron. Using the works of various authors, this paper explores the distribution of contractible edges in 3-connected graphs as needed to prove a major result originally by Tutte: each edge in a 3-connected graph is part of at least 2 peripheral cycles that share only the edge and its end vertices. A complete, alternative proof of this theorem is provided. The inductive step is generalized into a new independent lemma, which states that each edge in a 3-connected graph with a non-adjacent contractible edge has at least as many peripheral cycles as in the contracted one.

Planarity and non-separating cycles in uniform high genus quadrangulations

We study large uniform random bipartite quadrangulations whose genus grows linearly with the number of faces. Their local convergence was recently established by Budzinski and the author [9, 10]. Here we study several properties of these objects which are not captured by the local topology. Namely we show that balls around the root are planar with high probability up to logarithmic radius, and we prove that there exist non-contractible cycles of constant length with positive probability.

Chromatic Number, Induced Cycles, and Non-separating Cycles

We study two parameters obtained from the Euler characteristic by replacing the number of faces with that of induced and induced non-separating cycles. By establishing monotonicity of such parameters under certain homomorphism and edge contraction, we obtain new upper bounds on the chromatic number in terms of the number of induced cycles and the Hadwiger number in terms of the number of induced non-separating cycles. As an application, we show that every 3-connected graph with average degree at least 2k for some $$k\ge 2$$ have at least $$(k-1)|V|+Ck^{3}\log ^{3/2}k$$ induced non-separating cycles for some explicit constant $$C>0$$ . This improves the previous best known lower bound $$(k-1)|V|+1$$ , which follows from Tutte’s cycle space theorem. We also give a short proof of this theorem of Tutte.

Cycle double covers and non-separating cycles

Which 2-regular subgraph R of a cubic graph G can be extended to a cycle double cover of G? We provide a condition which ensures that every R satisfying this condition is part of a cycle double cover of G. As one consequence, we prove that every 2-connected cubic graph which has a decomposition into a spanning tree and a 2-regular subgraph C consisting of k circuits with k≤3, has a cycle double cover containing C.

Finding shortest non-trivial cycles in directed graphs on surfaces

Let $D$ be a weighted directed graph cellularly embedded in a surface of genus $g$, orientable or not, possibly with boundary. We describe algorithms to compute shortest non-contractible and shortest surface non-separating cycles in $D$, generalizing previous results that dealt with undirected graphs. Our first algorithm computes such cycles in $O(n^2\log n)$ time, where $n$ is the total number of vertices and edges of $D$, thus matching the complexity of the best general algorithm in the undirected case. It revisits and extends Thomassen's 3-path condition; the technique applies to other families of cycles as well. We also provide more efficient algorithms in special cases, such as graphs with small genus or bounded treewidth, using a divide-and-conquer technique that simplifies the graph while preserving the topological properties of its cycles. Finally, we give an efficient output-sensitive algorithm, whose running time depends on the length of the shortest non-contractible or non-separating cycle.

Spanning trees with nonseparating paths

We consider questions related to the existence of spanning trees in connected graphs with the property that, after the removal of any path in the tree, the graph remains connected. We show that, for planar graphs, the existence of trees with this property is closely related to the Hamiltonicity of the graph. For graphs with a 1- or 2-vertex cut, the Hamiltonicity also plays a central role. We also deal with spanning trees satisfying this property restricted to paths arising from fundamental cycles. The cycle space of a graph can be generated by the fundamental cycles of every spanning tree, and Tutte showed that, for a 3-connected graph, it can be generated by nonseparating cycles. We are also interested in the existence of a fundamental basis consisting of nonseparating cycles.

Multiple-Source Shortest Paths in Embedded Graphs

Let $G$ be a directed graph with $n$ vertices and nonnegative weights in its directed edges, embedded on a surface of genus $g$, and let $f$ be an arbitrary face of $G$. We describe a randomized algorithm to preprocess the graph in $O(gn \log n)$ time with high probability, so that the shortest-path distance from any vertex on the boundary of $f$ to any other vertex in $G$ can be retrieved in $O(\log n)$ time. Our result directly generalizes the $O(n\log n)$-time algorithm of Klein [Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, 2005] for multiple-source shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortest-path tree as its source point moves continuously around the boundary of $f$. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in $O(g^2 n\log n)$ time with high probability. Our high-probability time bounds hold in the worst case for generic edge weights or with an additional $O(\log n)$ factor for arbitrary edge weights.

Algorithms for the edge-width of an embedded graph

Let G be an unweighted graph of complexity n embedded in a surface of genus g, orientable or not. We describe improved algorithms to compute a shortest non-contractible and a shortest non-separating cycle in G.If k is an integer, we can compute such a non-trivial cycle with length at most k in O(gnk) time, or correctly report that no such cycle exists. In particular, on a fixed surface, we can test in linear time whether the edge-width or face-width of a graph is bounded from above by a constant. This also implies an output-sensitive algorithm to compute a shortest non-trivial cycle that runs in O(gnk0) time, where k0 is the length of the cycle.We also give an approximation algorithm for the shortest non-trivial cycle. If a parameter 0<ε<1 is given, we compute in O(gn/ε) time a non-trivial cycle whose length is at most 1+ε times the length of the shortest non-trivial cycle.

Finding one tight cycle

A cycle on a combinatorial surface is tight if it as short as possible in its (free) homotopy class. We describe an algorithm to compute a single tight, noncontractible, essentially simple cycle on a given orientable combinatorial surface in O ( n log n ) time. The only method previously known for this problem was to compute the globally shortest noncontractible or nonseparating cycle in O (min{ g 3 , n }, n log n ) time, where g is the genus of the surface. As a consequence, we can compute the shortest cycle freely homotopic to a chosen boundary cycle in O ( n log n ) time, a tight octagonal decomposition in O ( gn log n ) time, and a shortest contractible cycle enclosing a nonempty set of faces in O ( n log 2 n ) time.

Note on non-separating and removable cycles in highly connected graphs

We combine two well-known results by Mader and Thomassen, respectively. Namely, we prove that for any k -connected graph G ( k ≥ 4 ), there is an induced cycle C such that G − V ( C ) is ( k − 3 ) -connected and G − E ( C ) is ( k − 2 ) -connected. Both “ ( k − 3 ) -connected” and “ ( k − 2 ) -connected” are best possible in a sense.

Nonseparating Induced Cycles Consisting of Contractible Edges in k-Connected Graphs

Egawa and Saito proved that every k-connected graph with girth at least 4 has an induced cycle C such that $G-V(C)$ is $(k-3)$-connected, and every edge of C is contractible. This means that we can find not only a nonseparating cycle C but also one that consists of contractible edges. Motivated by this result, we prove that if G is a k-connected graph which does not contain $K_4^{-}$, then G has an induced cycle C such that $G - V(C)$ is $(k-2)$-connected and either every edge of C is k-contractible or C is a triangle. As a corollary of this result, we get the following result: Every k-connected graph with girth at least 4 has an induced cycle C such that $G-V(C)$ is $(k-2)$-connected, and every edge of C is contractible. This theorem is a generalization of some known theorems. In particular, this generalizes the above-mentioned result proved by Egawa and Saito and the result of Egawa which says that a k-connected graph with girth at least 4 has an induced cycle C such that $G-V(C)$ is $(k-2)$-connected.

Minimum cycle bases of graphs on surfaces

In this paper we study the cycle base structures of embedded graphs on surfaces. We first give a sufficient and necessary condition for a set of facial cycles to be contained in a minimum cycle base (or MCB in short) and then set up a 1–1 correspondence between the set of MCBs and the set of collections of nonseparating cycles which are in general positions on surfaces and are of shortest total length. This provides a way to enumerate MCBs in a graph via nonseparating cycles. In particular, some known results such as P.F. Stadler's work on Halin graphs [Minimum cycle bases of Halin graphs, J. Graph Theory 43 (2003) 150–155] and Leydold and Stadler's results on outer-planar graphs [Minimum cycle bases of outerplanar graphs, Electronic J. Combin. 5(16) (1998) 14] are concluded. As applications, the number of MCBs in some types of graphs embedded in lower surfaces (with arbitrarily high genera) is found. Finally, we present an interpolation theorem for the number of one-sided cycles contained in MCB of an embedded graph.

Finding Shortest Non-Separating and Non-Contractible Cycles for Topologically Embedded Graphs

We present an algorithm for finding shortest surface non-separating cycles in graphs embedded on surfaces in \(O(g^{3/2}V^{3/2}\log V+g^{5/2}V^{1/2})\) time, where V is the number of vertices in the graph and g is the genus of the surface. If \(g=o(V^{1/3})\), this represents an improvement over previous results by Thomassen, and Erickson and Har-Peled. We also give algorithms to find a shortest non-contractible cycle in \(O(g^{O(g)}V^{3/2})\) time, which improves previous results for fixed genus. This result can be applied for computing the face-width and the non-separating face-width of embedded graphs. Using similar ideas we provide the first near-linear running time algorithm for computing the face-width of a graph embedded on the projective plane, and an algorithm to find the face-width of embedded toroidal graphs in \(O(V^{5/4}\log V)\) time.

Removing excess topology from isosurfaces

Many high-resolution surfaces are created through isosurface extraction from volumetric representations, obtained by 3D photography, CT, or MRI. Noise inherent in the acquisition process can lead to geometrical and topological errors. Reducing geometrical errors during reconstruction is well studied. However, isosurfaces often contain many topological errors in the form of tiny handles. These nearly invisible artifacts hinder subsequent operations like mesh simplification, remeshing, and parametrization. In this article we present a practical method for removing handles in an isosurface. Our algorithm makes an axis-aligned sweep through the volume to locate handles, compute their sizes, and selectively remove them. The algorithm is designed to facilitate out-of-core execution. It finds the handles by incrementally constructing and analyzing a Reeb graph. The size of a handle is measured by a short nonseparating cycle. Handles are removed robustly by modifying the volume rather than attempting "mesh surgery." Finally, the volumetric modifications are spatially localized to preserve geometrical detail. We demonstrate topology simplification on several complex models, and show its benefits for subsequent surface processing.

Extending precolorings of subgraphs of locally planar graphs

The width of an embedded graph is the length of a shortest noncontractible cycle. Suppose G is embedded in a surface S (either orientable or not) with large width. In this case G is said to be locally planar. Suppose P⊂ V( G) is a set of vertices such that the components of G[ P] are each 2-colorable, have bounded diameter and are suitably distant from each other. We show that any 5-coloring of G[ P] in which each component is 2-colored extends to a 5-coloring of all of G. Thus, for an arbitrary surface, the extension theorems for precolorings of subgraphs of locally planar graphs parallel the results for planar graphs. Crucial to the proof of this result is the nice cycle lemma, viz. If C is a minimal, noncontractible, and nonseparating cycle in a so-called orderly triangulation of at least moderate width, then there is a cycle C′ such that C′ lies within the fourth neighborhood of C, C′ is minimal, homotopic to C, and C′ either has even length or contains a vertex of degree 4. Such a nice cycle is useful in producing 5-colorings. We introduce the idea of optimal shortcuts in order to prove the nice cycle lemma and the idea of relative width in order to prove the main theorem. Our results generalize to extension theorems for precolorings with q≥3 colors.

Connected Hyperplanes in Binary Matroids

In this paper, we prove that any simple and cosimple connected binary matroid has at least four connected hyperplanes. We further prove that each element in such a matroid is contained in at least two connected hyperplanes. Our main result generalizes a matroid result of Kelmans, and independently, of Seymour. The following consequence of the main result generalizes a graph result of Thomassen and Toft on induced non-separating cycles and another graph result of Kaugars on deletable vertices. If G is a simple 2-connected graph with minimum degree at least 3, then, for every edge e , there are at least two induced non-separating cycles avoiding e and two deletable vertices non-incident to e . Moreover, G has at least four induced non-separating cycles.

Separating and Nonseparating Disjoint Homotopic Cycles in Graph Embeddings

We show that if a graph G is embedded in a surface Σ with representativity ρ , then G contains at least ⌊( ρ −1)/2⌋ pairwise disjoint, pairwise homotopic, nonseparating (in Σ ) cycles, and G contains at least ⌊( ρ −1)/8⌋−1 pairwise disjoint, pairwise homotopic, separating, noncontractible cycles.

A charcterization of 4‐edge‐connected eulerian graphs

AbstractLet v be an arbitrary vertex of a 4‐edge‐connected Eulerian graph G. First we show the existence of a nonseparating cycle decompositiion of G with respect to v. With the help of this decomposition we are then able to construct 4 edge‐independent spanning trees with the common root v in the sam graph. We conclude that an Eulerian graph G is 4‐edge‐connected iff for every vertex r ϵ V (G) there exist 4 edge‐independent spanning trees with a common root r. © 1996 John Wiley &amp; Sons, Inc.

Non-separating cycles and discrete Jordan curves

We study induced non-separating cycles in 2-connected and 3-connected graphs. As consequences, two conjectures of Vince and Little on discrete Jordan curves are solved, in particular, they are true for simple 2-connected graphs with minimum degree at least six (this is best possible). We also disprove a conjecture of Thomassen and Toft about the structure of 2-connected graphs in which all induced cycles are separating.