arXiv subjects

Jens M. Schmidt

Publications and source records attributed to Jens M. Schmidt.

At least 19 recordsLinked to original sources

arXiv2024

Trees and co-trees in planar 3-connected graphs An easier proof via Schnyder woods

Let $G$ be a 3-connected planar graph. Define the co-tree of a spanning tree $T$ of $G$ as the graph induced by the dual edges of $E(G)-E(T)$. The well-known cut-cycle duality implies that the co-tree is itself a tree. Let a $k$-tree be a spanning tree with maximum degree $k$. In 1970, Grünbaum conjectured that every 3-connected planar graph contains a 3-tree whose co-tree is also a 3-tree. In 2014, Biedl showed that every such graph contains a 5-tree whose co-tree is a 5-tree. In this paper, we present an easier proof of Biedl's result

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arXiv2024

Toward Grünbaum's Conjecture

Given a spanning tree $T$ of a planar graph $G$, the co-tree of $T$ is the spanning tree of the dual graph $G^*$ with edge set $(E(G)-E(T))^*$. Grünbaum conjectured in 1970 that every planar 3-connected graph $G$ contains a spanning tree $T$ such that both $T$ and its co-tree have maximum degree at most 3. While Grünbaum's conjecture remains open, Biedl proved that there is a spanning tree $T$ such that $T$ and its co-tree have maximum degree at most 5. By using new structural insights into Schnyder woods, we prove that there is a spanning tree $T$ such that $T$ and its co-tree have maximum degree at most 4.

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arXiv2023

Rooted Minors and Locally Spanning Subgraphs

Results on the existence of various types of spanning subgraphs of graphs are milestones in structural graph theory and have been diversified in several directions. In the present paper, we consider "local" versions of such statements. In 1966, for instance, D. W. Barnette proved that a $3$-connected planar graph contains a spanning tree of maximum degree at most $3$. A local translation of this statement is that if $G$ is a planar graph, $X$ is a subset of specified vertices of $G$ such that $X$ cannot be separated in $G$ by removing $2$ or fewer vertices of $G$, then $G$ has a tree of maximum degree at most $3$ containing all vertices of $X$. Our results constitute a general machinery for strengthening statements about $k$-connected graphs (for $1 \leq k \leq 4$) to locally spanning versions, i.e. subgraphs containing a set $X\subseteq V(G)$ of a (not necessarily planar) graph $G$ in which only $X$ has high connectedness. Given a graph $G$ and $X\subseteq V(G)$, we say $M$ is a minor of $G$ rooted at $X$, if $M$ is a minor of $G$ such that each bag of $M$ contains at most one vertex of $X$ and $X$ is a subset of the union of all bags. We show that $G$ has a highly connected minor rooted at $X$ if $X\subseteq V(G)$ cannot be separated in $G$ by removing a few vertices of $G$. Combining these investigations and the theory of Tutte paths in the planar case yields to locally spanning versions of six well-known results about degree-bounded trees, hamiltonian paths and cycles, and $2$-connected subgraphs of graphs.

math.CO↗

arXiv2021

Circumference of essentially 4-connected planar triangulations

A $3$-connected graph $G$ is essentially $4$-connected if, for any $3$-cut $S\subseteq V(G)$ of $G$, at most one component of $G-S$ contains at least two vertices. We prove that every essentially $4$-connected maximal planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2}{3}(n+4)$; moreover, this bound is sharp.

math.CO↗

arXiv2020

Dynamics of Cycles in Polyhedra I: The Isolation Lemma

A cycle $C$ of a graph $G$ is \emph{isolating} if every component of $G-V(C)$ is a single vertex. We show that isolating cycles in polyhedral graphs can be extended to larger ones: every isolating cycle $C$ of length $6 \leq |E(C)| < \left \lfloor \frac{2}{3}(|V(G)|+4) \right \rfloor$ implies an isolating cycle $C'$ of larger length that contains $V(C)$. By "hopping" iteratively to such larger cycles, we obtain a powerful and very general inductive motor for proving long cycles and computing them (we will give an algorithm with quadratic running time). This is the first step towards the so far elusive quest of finding a universal induction that captures longest cycles of polyhedral graph classes. Our motor provides also a method to prove linear lower bounds on the length of Tutte cycles, as $C'$ will be a Tutte cycle of $G$ if $C$ is. We prove in addition that $|E(C')| \leq |E(C)|+3$ if $G$ contains no face of size five, which gives a new tool for results about cycle spectra, and provides evidence that faces of size five may obstruct long cycles in many graph classes. We test our motor on the following conjecture about essentially 4-connected graphs. A planar graph is \emph{essentially $4$-connected} if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Jackson and Wormald proved that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2}{5}(n+2)$, and this result has recently been improved multiple times, culminating in the lower bound $\frac{5}{8}(n+2)$. However, the currently best known upper bound is given by an infinite family of such graphs in which no graph $G$ contains a cycle that is longer than $\left \lfloor \frac{2}{3}(n+4) \right \rfloor$; this upper bound is still unmatched. Using isolating cycles, we improve the lower bound to match the upper. All our results are tight.

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arXiv2019

On the Circumference of Essentially 4-connected Planar Graphs

A planar graph is essentially $4$-connected if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Jackson and Wormald proved that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2n+4}{5}$, and this result has recently been improved multiple times. In this paper, we prove that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{5}{8}(n+2)$. This improves the previously best-known lower bound $\frac{3}{5}(n+2)$.

math.CO↗

arXiv2019

Compact Cactus Representations of all Non-Trivial Min-Cuts

Recently, Kawarabayashi and Thorup presented the first deterministic edge-connectivity recognition algorithm in near-linear time. A crucial step in their algorithm uses the existence of vertex subsets of a simple graph $G$ on $n$ vertices whose contractions leave a multigraph with $\tilde{O}(n/δ)$ vertices and $\tilde{O}(n)$ edges that preserves all non-trivial min-cuts of $G$, where $δ$ is the minimum degree of $G$ and $\tilde{O}$ hides logarithmic factors. We present a simple argument that improves this contraction-based sparsifier by eliminating the poly-logarithmic factors, that is, we show a contraction-based sparsification that leaves $O(n/δ)$ vertices and $O(n)$ edges, preserves all non-trivial min-cuts and can be computed in near-linear time $\tilde{O}(m)$, where $m$ is the number of edges of $G$. We also obtain that every simple graph has $O((n/δ)^2)$ non-trivial min-cuts. Our approach allows to represent all non-trivial min-cuts of a graph by a cactus representation, whose cactus graph has $O(n/δ)$ vertices. Moreover, this cactus representation can be derived directly from the standard cactus representation of all min-cuts in linear time. We apply this compact structure to show that all min-cuts can be explicitly listed in $\tilde{O}(m) + O(n^2 / δ)$ time for every simple graph, which improves the previous best time bound $O(nm)$ given by Gusfield and Naor.

math.CO↗

arXiv2017

Longer Cycles in Essentially 4-Connected Planar Graphs

A planar 3-connected graph $G$ is called \emph{essentially $4$-connected} if, for every 3-separator $S$, at least one of the two components of $G-S$ is an isolated vertex. Jackson and Wormald proved that the length $\mathop{\rm circ}\nolimits(G)$ of a longest cycle of any essentially 4-connected planar graph $G$ on $n$ vertices is at least $\frac{2n+4}{5}$ and Fabrici, Harant and Jendrol' improved this result to $\mathop{\rm circ}\nolimits(G)\geq \frac{1}{2}(n+4)$. In the present paper, we prove that an essentially 4-connected planar graph on $n$ vertices contains a cycle of length at least $\frac{3}{5}(n+2)$ and that such a cycle can be found in time $O(n^2)$.

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arXiv2017

Computing Tutte Paths

Tutte paths are one of the most successful tools for attacking Hamiltonicity problems in planar graphs. Unfortunately, results based on them are non-constructive, as their proofs inherently use an induction on overlapping subgraphs and these overlaps hinder to bound the running time to a polynomial. For special cases however, computational results of Tutte paths are known: For 4-connected planar graphs, Tutte paths are in fact Hamiltonian paths and Chiba and Nishizeki showed how to compute such paths in linear time. For 3-connected planar graphs, Tutte paths have a more complicated structure, and it has only recently been shown that they can be computed in polynomial time. However, Tutte paths are defined for general 2-connected planar graphs and this is what most applications need. Unfortunately, no computational results are known. We give the first efficient algorithm that computes a Tutte path (for the general case of 2-connected planar graphs). One of the strongest existence results about such Tutte paths is due to Sanders, which allows to prescribe the end vertices and an intermediate edge of the desired path. Encompassing and strengthening all previous computational results on Tutte paths, we show how to compute this special Tutte path efficiently. Our method refines both, the results of Thomassen and Sanders, and avoids overlapping subgraphs by using a novel iterative decomposition along 2-separators. Finally, we show that our algorithm runs in quadratic time.

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arXiv2017

Cut Tree Structures with Applications on Contraction-Based Sparsification

We introduce three new cut tree structures of graphs $G$ in which the vertex set of the tree is a partition of $V(G)$ and contractions of tree vertices satisfy sparsification requirements that preserve various types of cuts. Recently, Kawarabayashi and Thorup \cite{Kawarabayashi2015a} presented the first deterministic near-linear edge-connectivity recognition algorithm. A crucial step in this algorithm uses the existence of vertex subsets of a simple graph $G$ whose contractions leave a graph with $\tilde{O}(n/δ)$ vertices and $\tilde{O}(n)$ edges ($n := |V(G)|$) such that all non-trivial min-cuts of $G$ are preserved. We improve this result by eliminating the poly-logarithmic factors, that is, we show a contraction-based sparsification that leaves $O(n/δ)$ vertices and $O(n)$ edges and preserves all non-trivial min-cuts. We complement this result by giving a sparsification that leaves $O(n/δ)$ vertices and $O(n)$ edges such that all (possibly not minimum) cuts of size less than $δ$ are preserved, by using contractions in a second tree structure. As consequence, we have that every simple graph has $O(n/δ)$ $δ$-edge-connected components, and, if it is connected, it has $O((n/δ)^2)$ non-trivial min-cuts. All these results are proven to be asymptotically optimal. By using a third tree structure, we give a new lower bound on the number of \emph{pendant pairs}. The previous best bound was given 1974 by Mader, who showed that every simple graph contains $Ω(δ^2)$ pendant pairs. We improve this result by showing that every simple graph $G$ with $δ\geq 5$ or $λ\geq 4$ or $κ\geq 3$ contains $Ω(δn)$ pendant pairs. We prove that this bound is asymptotically tight from several perspectives, and that $Ω(δn)$ pendant pairs can be computed efficiently.

math.CO↗

arXiv2016

More on foxes

An edge in a $k$-connected graph $G$ is called {\em $k$-contractible} if the graph $G/e$ obtained from $G$ by contracting $e$ is $k$-connected. Generalizing earlier results on $3$-contractible edges in spanning trees of $3$-connected graphs, we prove that (except for the graphs $K_{k+1}$ if $k \in \{1,2\}$) (a) every spanning tree of a $k$-connected triangle free graph has two $k$-contractible edges, (b) every spanning tree of a $k$-connected graph of minimum degree at least $\frac{3}{2}k-1$ has two $k$-contractible edges, (c) for $k>3$, every DFS tree of a $k$-connected graph of minimum degree at least $\frac{3}{2}k-\frac{3}{2}$ has two $k$-contractible edges, (d) every spanning tree of a cubic $3$-connected graph nonisomorphic to $K_4$ has at least $\frac{1}{3}|V(G)|-1$ many $3$-contractible edges, and (e) every DFS tree of a $3$-connected graph nonisomorphic to $K_4$, the prism, or the prism plus a single edge has two 3-contractible edges. We also discuss in which sense these theorems are best possible.

math.CO↗

arXiv2016

Computing Vertex-Disjoint Paths using MAOs

Let G be a graph with minimum degree $δ$. It is well-known that maximal adjacency orderings (MAOs) compute a vertex set S such that every pair of S is connected by at least $δ$ internally vertex-disjoint paths in G. We present an algorithm that, given any pair of S, computes these $δ$ paths in linear time O(n+m). This improves the previously best solutions for these special vertex pairs, which were flow-based. Our algorithm simplifies a proof about pendant pairs of Mader and makes a purely existential proof of Nagamochi algorithmic.

cs.DM↗

arXiv2016

Mondshein Sequences (a.k.a. (2,1)-Orders)

Canonical orderings [STOC'88, FOCS'92] have been used as a key tool in graph drawing, graph encoding and visibility representations for the last decades. We study a far-reaching generalization of canonical orderings to non-planar graphs that was published by Lee Mondshein in a PhD-thesis at M.I.T. as early as 1971. Mondshein proposed to order the vertices of a graph in a sequence such that, for any i, the vertices from 1 to i induce essentially a 2-connected graph while the remaining vertices from i+1 to n induce a connected graph. Mondshein's sequence generalizes canonical orderings and became later and independently known under the name non-separating ear decomposition. Surprisingly, this fundamental link between canonical orderings and non-separating ear decomposition has not been established before. Currently, the fastest known algorithm for computing a Mondshein sequence achieves a running time of O(nm); the main open problem in Mondshein's and follow-up work is to improve this running time to subquadratic time. After putting Mondshein's work into context, we present an algorithm that computes a Mondshein sequence in optimal time and space O(m). This improves the previous best running time by a factor of n. We illustrate the impact of this result by deducing linear-time algorithms for five other problems, for four out of which the previous best running times have been quadratic. In particular, we show how to - compute three independent spanning trees of a 3-connected graph in time O(m), - improve the preprocessing time from O(n^2) to O(m) for a data structure reporting 3 internally disjoint paths between any given vertex pair, - derive a very simple O(n)-time planarity test once a Mondshein sequence has been computed, - compute a nested family of contractible subgraphs of 3-connected graphs in time O(m), - compute a 3-partition in time O(m).

cs.DS↗

arXiv2016

Edge-Orders

Canonical orderings and their relatives such as st-numberings have been used as a key tool in algorithmic graph theory for the last decades. Recently, a unifying concept behind all these orders has been shown: they can be described by a graph decomposition into parts that have a prescribed vertex-connectivity. Despite extensive interest in canonical orderings, no analogue of this unifying concept is known for edge-connectivity. In this paper, we establish such a concept named edge-orders and show how to compute (1,1)-edge-orders of 2-edge-connected graphs as well as (2,1)-edge-orders of 3-edge-connected graphs in linear time, respectively. While the former can be seen as the edge-variants of st-numberings, the latter are the edge-variants of Mondshein sequences and non-separating ear decompositions. The methods that we use for obtaining such edge-orders differ considerably in almost all details from the ones used for their vertex-counterparts, as different graph-theoretic constructions are used in the inductive proof and standard reductions from edge- to vertex-connectivity are bound to fail. As a first application, we consider the famous Edge-Independent Spanning Tree Conjecture, which asserts that every k-edge-connected graph contains k rooted spanning trees that are pairwise edge-independent. We illustrate the impact of the above edge-orders by deducing algorithms that construct 2- and 3-edge independent spanning trees of 2- and 3-edge-connected graphs, the latter of which improves the best known running time from O(n^2) to linear time.

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arXiv2016

A Tight Bound for Minimal Connectivity

For minimally $k$-connected graphs on $n$ vertices, Mader proved a tight lower bound for the number $|V_k|$ of vertices of degree $k$ in dependence on $n$ and $k$. Oxley observed 1981 that in many cases a considerably better bound can be given if $m := |E|$ is used as additional parameter, i.e. in dependence on $m$, $n$ and $k$. It was left open to determine whether Oxley's bound is best possible. We show that this is not the case, but propose a closely related bound that deviates from Oxley's long-standing one only for small values of $m$. We prove that this new bound is best possible. The bound contains Mader's bound as special case.

math.CO↗

arXiv2015

Small-Area Orthogonal Drawings of 3-Connected Graphs

It is well-known that every graph with maximum degree 4 has an orthogonal drawing with area at most $\frac{49}{64} n^2+O(n) \approx 0.76n^2$. In this paper, we show that if the graph is 3-connected, then the area can be reduced even further to $\frac{9}{16}n^2+O(n) \approx 0.56n^2$. The drawing uses the 3-canonical order for (not necessarily planar) 3-connected graphs, which is a special Mondshein sequence and can hence be computed in linear time. To our knowledge, this is the first application of a Mondshein sequence in graph drawing.

cs.CG↗

arXiv2015

Certifying 3-Edge-Connectivity

We present a certifying algorithm that tests graphs for 3-edge-connectivity; the algorithm works in linear time. If the input graph is not 3-edge-connected, the algorithm returns a 2-edge-cut. If it is 3-edge-connected, it returns a construction sequence that constructs the input graph from the graph with two vertices and three parallel edges using only operations that (obviously) preserve 3-edge-connectivity. Additionally, we show how compute and certify the 3-edge-connected components and a cactus representation of the 2-cuts in linear time. For 3-vertex-connectivity, we show how to compute the 3-vertex-connected components of a 2-connected graph.

cs.DS↗

arXiv2015

Counting K_4-Subdivisions

A fundamental theorem in graph theory states that any 3-connected graph contains a subdivision of $K_4$. As a generalization, we ask for the minimum number of $K_4$-subdivisions that are contained in every $3$-connected graph on $n$ vertices. We prove that there are $Ω(n^3)$ such $K_4$-subdivisions and show that the order of this bound is tight for infinitely many graphs. We further investigate a better bound in dependence on $m$ and prove that the computational complexity of the problem of counting the exact number of $K_4$-subdivisions is $\#P$-hard.

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