A new local structure theorem for annotated graphs of bounded $\mathsf{depth}_2$ with a spanning analogue of the celebrated Vital Linkage Theorem and matching lower bounds showing that beyond bounded $\mathsf{depth}_2$ no irrelevant-vertex rule can exist, even on planar graphs.
Abstract
The Irrelevant Vertex Technique is one of the cornerstones of algorithmic graph theory, underlying Robertson and Seymour's algorithm for \textsc{Disjoint Paths} and much of the algorithmic Graph Minors theory. We show that, in the setting of spanning routing, this technique exhibits an exact combinatorial limitation. Unlike classical routing problems, spanning routing is not governed by the number of distinguished vertices but by the way they are distributed throughout the graph. The input is a triple $(G,R,\mathcal{T})$ where $(G,R)$ is an annotated graph and $\mathcal{T}$ is a set of terminal pairs. The goal is to determine if $G$ contains a family of internally disjoint paths connecting the pairs in $\mathcal{T}$ such that the union of the paths spans the set $R$. We identify a new structural parameter of annotated graphs, called $\mathsf{depth}_2$, that measures precisely this phenomenon. Our main result is a complete combinatorial dichotomy: for every red-minor-closed class of annotated graphs, the Irrelevant Vertex Technique applies to \textsc{Spanning Disjoint Paths} \textsl{if and only if} the class has bounded $\mathsf{depth}_2$. Thus $\mathsf{depth}_2$ forms the exact structural boundary between classes where the Robertson-Seymour paradigm survives and those where it breaks down. Our proof combines a new local structure theorem for annotated graphs of bounded $\mathsf{depth}_2$ with a spanning analogue of the celebrated Vital Linkage Theorem. The resulting algorithm solves \textsc{Spanning Disjoint Paths} in time $2^{2^{\mathbf{poly}(k+d)}}\cdot n^2$ where $d$ is the $\mathsf{depth}_2$ of the input instance. We provide matching lower bounds showing that beyond bounded $\mathsf{depth}_2$ no irrelevant-vertex rule can exist, even on planar graphs. In particular, $\mathsf{depth}_2$ is the exact combinatorial barrier for the Irrelevant Vertex Technique under spanning constraints.
The vertex arboricity $\mathrm{va}(G)$ of a multigraph $G$ is the minimum number $k$ for which $V(G)$ can be partitioned into $k$ subsets, each of which induces an acyclic subgraph of $G$. By definition, if $\mathrm{va}(G)= k$, then the chromatic number, $\chi(G)$, satisfies $k\leq \chi(G)\leq 2k$. Fundamental results by Borodin from 1976 and Bollob\'as and Manvel from 1979 imply an analog of Gallai's lower bound on the number of edges in a $(2k-1)$-critical graph. We consider a slight generalization of vertex arboricity in the setting of DP-coloring. Using this framework, we derive lower bounds on the number of edges in graphs critical for vertex arboricity and for list arboricity that are better than Gallai's bound, along with similar bounds in our DP-setting.
Peter Bradshaw, Alexandr V. Kostochka, Zimu Xiang· 0 citations
In the replacement paths (RP) problem, we are given a graph $G = (V, E)$ with $n = |V|$ and $m = |E|$, together with two vertices $s, t \in V$, and are asked to compute the shortest-path distance from $s$ to $t$ in $G \setminus e$ for every failed edge $e \in E$. The multiple source replacement paths (MSRP) problem is its natural generalization: given a set $S \subseteq V$ of $\sigma$ sources, compute the replacement path distances for all pairs in $S \times V$. In this paper, we present a randomized combinatorial algorithm that solves MSRP on unweighted directed graphs in $\tilde{O}(m\sqrt{\sigma n} + \sigma n^2)$ time, with all the output distances correct with high probability. This improves the best known bound $\tilde{O}(m\min\{\sigma\sqrt{n}, n\} + \sigma n^2)$ for directed graphs, which is obtained either by running the single source RP algorithm of Chechik and Magen [ICALP'20] from each source separately or by constructing and querying the all-pairs distance sensitivity oracle of Bernstein and Karger [STOC'09]. Our running time is essentially tight among combinatorial algorithms because Gupta, Jain, and Modi [PODC'20] proved a lower bound of $m{(\sigma n)}^{1/2-o(1)}$ for such algorithms, which holds even on undirected graphs, and the additive term $\sigma n^2$ is proportional to the time needed to write down the $\Theta(\sigma n^2)$ output distances. The algorithm is also remarkably simple.
It is shown that bidirectional Dijkstra is still instance-optimal on simple undirected weighted graphs under the order-oblivious model, where incident edges are given in a random order, and under the order-dependent model, where bidirectional Dijkstra is not instance-optimal.
Christian Bertram, Mads Vestergaard Jensen, Mikkel Thorup et al.· 0 citations
A {\it threshold graph} is a graph that can be constructed from the one-vertex graph by repeatedly adding either a dominating vertex or an isolated vertex. Motivated by an induced Ramsey-type problem for this class, we define $r'_2(s)$ to be the minimum integer $n$ such that every $n$-vertex graph contains an induced threshold graph on $s$ vertices. We establish exponential upper and lower bounds for $r'_2(s)$ and determine its exact values for $s\in\{3,4,5,6\}$. To study this problem from an edge-coloring perspective, we use the notion of an orderable coloring, introduced by Richer [{\it J. Combin. Theory Ser. B}, 80(1) (2000), 172--177]. An edge-colored graph is {\it orderable} if its vertices can be ordered so that, for each vertex, all edges from it to later vertices have the same color. Equivalently, $r'_2(s)$ is the minimum $n$ such that every $2$-edge-coloring of $K_n$ contains an orderable $K_s$. We also determine the exact value of the unordered canonical Ramsey number $CR(s, 3)$ for all $s \ge 3$, where $CR(s,3)$ denotes the minimum integer $n$ such that every edge-coloring of $K_n$ contains either an orderable $K_s$ or a rainbow $K_3$. More generally, for graphs $G$ and $H$, we study $r'_2(G)$, the corresponding $2$-color Ramsey number for an orderable $G$, and $CR(G,H)$, where the alternative is a rainbow $H$. For complete bipartite graphs, we prove that for every fixed $s$, $r'_2(K_{s,t}) = CR(K_{s,t}, K_3)= \left(\frac{2^s}{s+1}+o(1)\right)t$ as $t\to\infty$. For $s\in \{2,3\}$, we further determine the exact values of these parameters for infinitely many $t$, using constructions arising from strongly regular graphs, Hadamard matrices and conference matrices.
The $\text{bk}(G)$ of a graph $G$ is the maximum number of triangles sharing a common edge. Motivated by a classical conjecture of Erd\H{o}s, spectral lower bounds for the booksize have received considerable attention. For a positive divisor $s$ of $m-1$ with $\frac{m-1}{s}\ge2$, let $S_{m,s}^{+}$ be obtained from $K_{s,\frac{m-1}{s}}$ by adding one edge inside the part of order $\frac{m-1}{s}$. Zhai et al. proved that, apart from this explicit family, every $m$-edge non-bipartite graph satisfying $\rho(G)^2\ge m-1+\frac{2}{\rho(G)-1}$ has booksize greater than $\frac{1}{240}\sqrt{m}$, and they asked for the best possible constant. We answer this question asymptotically. For every $0<\varepsilon<\frac{1}{4}$ and all sufficiently large $m$, every $m$-edge non-bipartite graph $G$ without isolated vertices satisfying the same spectral condition either is isomorphic to $S_{m,s}^{+}$ for some such integer $s$, or satisfies $\text{bk}(G)>\left(\frac{1}{4}-\varepsilon\right)\sqrt{m}$. We also give infinitely many graphs outside the exceptional family showing that no constant larger than $\frac{1}{4}$ is possible. Thus $\frac{1}{4}$ is the optimal asymptotic constant in the problem of Zhai et al.
Starting from the stability theorem of Erd\H{o}s and Simonovits, stability problems for graphs forbidding a fixed subgraph have been studied in terms of edge numbers, spectral radii and subgraph counts. Let $\mathcal{N}(F,G)$ denote the number of unlabeled copies of $F$ in $G$. It is known that, for every fixed path $P_t$ and even cycle $C_{2a}$, the maximum number of copies in an $n$-vertex $C_{2\ell+1}$-free graph is attained by the bipartite Tur\'an graph $T_{n,2}$. In this paper we obtain strong structural stability for $C_{2\ell+1}$-free graphs in terms of copies of paths and even cycles. For fixed $\ell\ge2$ and $3\le r\le2\ell-1$, we show that if an $n$-vertex $C_{2\ell+1}$-free graph contains at least as many copies of $P_t$ or $C_{2a}$ as the corresponding suspended extremal construction, then it has the corresponding suspension structure. This gives exact high-chromatic extremal theorems for paths and even cycles. We also prove a counting theorem for nearly complete bipartite graphs. It shows that, for every fixed matching-admissible connected bipartite graph $F$, both imbalance between the two parts and missing cross-edges decrease the number of copies of $F$ by a term with a specified main coefficient. This theorem is independent of the forbidden odd cycle and converts subgraph-count assumptions into the edge bounds needed for the structural theorem.
