Subtrees search, cycle spectra and edge-connectivity structures

In the first part of this thesis, we study subtrees of specified weight in a tree $T$ with vertex weights $c: V(T) \rightarrow \mathbb{N}$. We introduce an overload-discharge method, and discover that there always exists some subtree $S$ whose weight $c(S) := \sum_{v \in V(S)} c(v)$ is close to $\frac{c(T)}{2}$; the smaller the weight $c(T)$ of $T$ is, the smaller difference between $c(S)$ and $\frac{c(T)}{2}$ we can assure. We also show that such a subtree can be found efficiently, namely in linear time. With this tool we prove that every planar Hamiltonian graph $G = (V(G), E(G))$ with minimum degree $\delta \geq 4$ has a cycle of length $k$ for every $k \in \{\lfloor \frac{|V(G)|}{2} \rfloor, \dots, \lceil \frac{|V(G)|}{2} \rceil + 3\}$ with $3 \leq k \leq |V(G)|$. Such a cycle can be found in linear time if a Hamilton cycle of the graph is given. In the second part of the thesis, we present three cut trees of a graph, each of them giving insights into the edge-connectivity structure. All three cut trees have in common that they cover a given binary symmetric irreflexive relation on the vertex set of the graph, while generalizing Gomory-Hu trees. With these cut trees we show the following: (i) every simple graph $G$ with $\delta \geq 5$ or with edge-connectivity $\lambda \geq 4$ or with vertex-connectivity $\kappa \geq 3$ contains at least $\frac{1}{24}\delta |V(G)|$ pendant pairs, where a pair of vertices $\{v, w\}$ is pendant if $\lambda_G(v,w) = \min\{d_G(v), d_G(w)\}$; (ii) every simple graph $G$ satisfying $\delta > 0$ has $O(|V(G)|/\delta)$ $\delta$-edge-connected components, and there are only $O(|V(G)|)$ edges left if these components are contracted; (iii) given a simple graph $G$ satisfying $\delta > 0$, one can find some vertex subsets in near-linear time such that all non-trivial min-cuts are preserved, and $O(|V(G)|/\delta)$ vertices and $O(|V(G)|)$ edges remain when these vertex subsets are contracted.Im ersten Teil dieser Dissertation untersuchen wir Teilbäume eines Baumes $T$ mit vorgegebenen Knotengewichten $c: V(T) \rightarrow \mathbb{N}$. Wir führen eine Overload-Discharge-Methode ein, und zeigen, dass es immer einen Teilbaum $S$ gibt, dessen Gewicht $c(S) := \sum_ {v \in V (S)} c(v)$ nahe $\frac{c(T)}{2}$ liegt. Je kleiner das Gewicht $c(T)$ von $T$ ist, desto geringer ist dabei die Differenz zwischen $c(S)$ und $\frac{c(T)}{2}$, die wir sicherstellen können. Wir zeigen auch, dass ein solcher Teilbaum effizient, nämlich in Linearzeit, berechnet werden kann. Unter Ausnutzung dieser Methode beweisen wir, dass jeder planare hamiltonsche Graph $G = (V(G), E(G))$ mit Mindestgrad $\delta \geq 4$ einen Kreis der Länge $k$ für jedes $k \in \{\lfloor \frac{|V(G)|}{2} \rfloor, \dots, \lceil \frac{|V(G)|}{2} \rceil + 3\}$ mit $3 \leq k \leq |V (G)|$ enthält. Dieser kann in Linearzeit berechnet werden, falls ein Hamilton-Kreis des Graphen bekannt ist. Im zweiten Teil der Dissertation stellen wir drei Schnittbäume eines Graphen vor, von denen jeder Einblick in die Kantenzusammenhangsstruktur des Graphen gibt. Allen drei Schnittbäumen ist gemeinsam, dass sie eine bestimmte binäre symmetrische irreflexive Relation auf der Knotenmenge des Graphen überdecken; die Bäume können als Verallgemeinerungen von Gomory-Hu-Bäumen aufgefasst werden. Die Schnittbäume implizieren folgende Aussagen: (i) Jeder schlichte Graph $G$, der $\delta \geq 5$ oder Kantenzusammenhang $\lambda \geq 4$ oder Knotenzusammenhang $\kappa \geq 3$ erfüllt, enthält mindestens $\frac{1}{24} \delta |V(G)|$ zusammengehörige Paare, wobei ein Paar von Knoten $\{v, w \}$ zusammengehörig ist, falls $\lambda_G (v, w) = \min \{d_G(v), d_G(w)\}$ ist. (ii) Jeder schlichte Graph $G$ mit $\delta > 0$ hat $O(|V (G)| / \delta)$ $\delta$-kantenzusammenhängende Komponenten, und es verbleiben lediglich $O(|V (G)|)$ Kanten, wenn diese Komponenten kontrahiert werden. (iii) Für jeden schlichten Graphen $G$ mit $\delta > 0$ sind Knotenmengen derart effizient berechenbar, dass alle nicht trivialen minimalen Schnitte erhalten bleiben, und $O(|V(G)| / \delta)$ Knoten und $O(|V(G)|)$ Kanten verbleiben, wenn diese Knotenmengen kontrahiert werden

Counting cycles in planar triangulations

We investigate the minimum number of cycles of specified lengths in planar $n$-vertex triangulations $G$. It is proven that this number is $\Omega(n)$ for any cycle length at most $3 + \max \{ {\rm rad}(G^*), \lceil (\frac{n-3}{2})^{\log_32} \rceil \}$, where ${\rm rad}(G^*)$ denotes the radius of the triangulation's dual, which is at least logarithmic but can be linear in the order of the triangulation. We also show that there exist planar hamiltonian $n$-vertex triangulations containing $O(n)$ many $k$-cycles for any $k \in \{ \lceil n - \sqrt[5]{n} \rceil, \ldots, n \}$. Furthermore, we prove that planar 4-connected $n$-vertex triangulations contain $\Omega(n)$ many $k$-cycles for every $k \in \{ 3, \ldots, n \}$, and that, under certain additional conditions, they contain $\Omega(n^2)$ $k$-cycles for many values of $k$, including $n$

A cut tree representation for pendant pairs

Two vertices v and w of a graph G are called a pendant pair if the maximal number of edge-disjoint paths in G between them is precisely min{d(v),d(w)}, where d denotes the degree function. The importance of pendant pairs stems from the fact that they are the key ingredient in one of the simplest and most widely used algorithms for the minimum cut problem today. Mader showed 1974 that every simple graph with minimum degree delta contains Omega(delta^2) pendant pairs; this is the best bound known so far. We improve this result by showing that every simple graph G with minimum degree delta >= 5 or with edge-connectivity lambda >= 4 or with vertex-connectivity kappa >= 3 contains in fact Omega(delta |V|) pendant pairs. We prove that this bound is tight from several perspectives, and that Omega(delta |V|) pendant pairs can be computed efficiently, namely in linear time when a Gomory-Hu tree is given. Our method utilizes a new cut tree representation of graphs

Shortness coefficient of cyclically 4-edge-connected cubic graphs

Grünbaum and Malkevitch proved that the shortness coefficient of cyclically 4-edge-connected cubic planar graphs is at most 76/77. Recently, this was improved to 359/366 (< 52/53) and the question was raised whether this can be strengthened to 41/42, a natural bound inferred from one of the Faulkner-Younger graphs. We prove that the shortness coefficient of cyclically 4-edge-connected cubic planar graphs is at most 37/38 and that we also get the same value for cyclically 4-edge-connected cubic graphs of genus g for any prescribed genus g ≥ 0. We also show that 45/46 is an upper bound for the shortness coefficient of cyclically 4-edge-connected cubic graphs of genus g with face lengths bounded above by some constant larger than 22 for any prescribed g ≥ 0

Tight cycle spectrum gaps of cubic 3-connected toroidal graphs

This note complements results on cycle spectra of planar graphs by investigating the toroidal case. Let k > 3 be an integer and G a cubic graph polyhedrally embedded on the torus with circumference at least k. For k = 3, it follows from Euler's formula that G has some (facial) cycle of length in [3, 6]. For k = 4 or 5, we show that G contains a cycle whose length lies in the interval [k, 12]; and for k > 5, we show that the same holds for the interval [k, 2k + 4]. This is best possible for all k > 4. On the other hand, for any non-negative integer gamma there exist 2-connected cubic graphs G of genus gamma, arbitrarily large face-width, and with arbitrarily large gaps in their cycle spectrum. The behaviour of cycle spectrum gaps of graphs polyhedrally embedded on surfaces of genus greater than 1 remains largely unknown.(c) 2022 Elsevier B.V. All rights reserved