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Given an unweighted directed graph, a sparsifier is a nonnegative reweighting of original arcs that preserves every in-degree and out-degree exactly. In this work, we study singular-value sparsifiers, which are generalizations of spectral sparsifiers. Singular-value sparsifiers find a number of applications in theoretical computer science, such as maintaining directed cuts and flows, speeding up directed Laplacian solvers, and sparsifying dense weight matrices in neural networks. Strong singular-value (SV) approximation supports sparse representations of directed random walks with guarantees that remain stable under products, and controls bilinear error in the left and right Schur-complement metrics. We prove that every unweighted directed graph on $n$ vertices admits an $\varepsilon$-SV sparsifier with $O(n\log(2n)/\varepsilon^2)$ arcs, improving the $O(n\log^5 n/\varepsilon^2)$ support bound of Lau, Wang, and Zhou. The sparsifier is a nonnegative reweighting of original arcs and preserves every in-degree and out-degree exactly. We also give a randomized construction with $\tilde{O}(m+n)$ real-arithmetic running time, where $m$ is the input support size, and the output has the same support bound and rational weights of polynomial bit length.

We give the first inductive characterization of planar graphs of treewidth $k$ for each $k \geq 3$. Our definition generalizes the well-known inductive definition of planar $3$-trees and the recent definition of so-called planar quasi-$4$-trees [Förster, 2024]. As a key step, which may be of independent interest, we show that every planar triangulation $G$ of treewidth $k$ admits a tree decomposition of width $k$ where each adhesion set induces a cycle in $G$. As an application of planar quasi-$4$-trees, we show that planar graphs of treewidth at most $4$ have vertex arboricity at most $2$. That is, the vertex-set of every planar graph of treewidth at most $4$ can be partitioned into two sets that each induce a forest.

A strong conflict-free vertex connection coloring of a graph is a graph coloring for which, for every two distinct vertices of the graph, there exists a shortest path between them in which some color in said coloring appears only once. The strong conflict-free vertex connection number, denoted by $\operatorname{svcfc}(G)$, is the number of colors in the strong conflict-free vertex connection coloring with the smallest amount of colors out of all such colorings of $G$. We study the problem of deciding whether this number is at most some threshold together with the parameter twin cover. Our first result is that when we have given $(G,k)$ in which $k$ is some target number of colors and given some twin cover $X$ of $G$ which is of size $t$, there is a polynomial time reduction to an equivalent instance which has at most $\max\{2,t+k2^t\}$ vertices. This result allows us to give a kernel for Strong CFVC Number which is parameterized by $\operatorname{tc}(G)+k$, which implies that Strong CFVC Number is FPT w.r.t. the twin-cover number of $G$ so long as $k$ is fixed. Our second result is that if $G$ is a connected graph for which there exists a twin cover of size $t$, then $χ(G)\le \operatorname{svcfc}(G)\le χ(G)+t$. Our last result is an an exact expression for the chromatic number of graphs whose twin cover number is bounded. Namely, we show that for every proper coloring $\varphi$ of $G[X]$, the minimum number of colors needed to extend $\varphi$ to all of $G$ is $K_\varphi=\max_{S\subseteq X}(|\varphi(S)|+m(S))$, and hence $χ(G)=\min_{\varphi\text{ proper on }G[X]} K_\varphi$. Our results indicate that twin cover is a meaningful parameter in the context of strong conflict-free vertex-connection, and show that once we have a fixed twin cover, the only difficulty left is in some additive threshold above the chromatic number.