Cops and Robber game in higher-dimensional manifolds with spherical and Euclidean metric

The recently introduced variation of the game of cops and robber is played on geodesic spaces. In this paper we establish some general strategies for the players, in particular the generalized radial strategy and the covering space strategy. Those strategies are then applied to the game on the $n$-dimensional ball, the sphere, and the torus.Comment: 19 page

On Minimizing the Energy of a Spherical Graph Representation

Graph representations are the generalization of geometric graph drawings from the plane to higher dimensions. A method introduced by Tutte to optimize properties of graph drawings is to minimize their energy. We explore this minimization for spherical graph representations, where the vertices lie on a unit sphere such that the origin is their barycentre. We present a primal and dual semidefinite program which can be used to find such a spherical graph representation minimizing the energy. We denote the optimal value of this program by $\rho(G)$ for a given graph $G$. The value turns out to be related to the second largest eigenvalue of the adjacency matrix of $G$, which we denote by $\lambda_2$. We show that for $G$ regular, $\rho(G) \leq \frac{\lambda_{2}}{2} \cdot v(G)$, and that equality holds if and only if the $\lambda_{2}$ eigenspace contains a spherical 1-design. Moreover, if $G$ is a random $d$-regular graph, $\rho(G)=\left(\sqrt{(d-1)} +o(1)\right)\cdot v(G)$, asymptotically almost surely.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Reconfiguration of plane trees in convex geometric graphs

A non-crossing spanning tree of a set of points in the plane is a spanning tree whose edges pairwise do not cross. Avis and Fukuda in 1996 proved that there always exists a flip sequence of length at most $2n-4$ between any pair of non-crossing spanning trees (where $n$ denotes the number of points). Hernando et al. proved that the length of a minimal flip sequence can be of length at least $\frac 32 n$. Two recent results of Aichholzer et al. and Bousquet et al. improved the Avis and Fukuda upper bound by proving that there always exists a flip sequence of length respectively at most $2n - \log n$ and $2n - \sqrt{n}$. We improve the upper bound by a linear factor for the first time in 25 years by proving that there always exists a flip sequence between any pair of non-crossing spanning trees $T_1,T_2$ of length at most $c n$ where $c \approx 1.95$. Our result is actually stronger since we prove that, for any two trees $T_1,T_2$, there exists a flip sequence from $T_1$ to $T_2$ of length at most $c |T_1 \setminus T_2|$. We also improve the best lower bound in terms of the symmetric difference by proving that there exists a pair of trees $T_1,T_2$ such that a minimal flip sequence has length $\frac 53 |T_1 \setminus T_2|$, improving the lower bound of Hernando et al. by considering the symmetric difference instead of the number of vertices. We generalize this lower bound construction to non-crossing flips (where we close the gap between upper and lower bounds) and rotations

Maximum Independent Set when excluding an induced minor: K1+tK2K_1 + tK_2 and tC3⊎C4tC_3 \uplus C_4

Dallard, Milani\v{c}, and \v{S}torgel [arXiv '22] ask if for every class excluding a fixed planar graph $H$ as an induced minor, Maximum Independent Set can be solved in polynomial time, and show that this is indeed the case when $H$ is any planar complete bipartite graph, or the 5-vertex clique minus one edge, or minus two disjoint edges. A positive answer would constitute a far-reaching generalization of the state-of-the-art, when we currently do not know if a polynomial-time algorithm exists when $H$ is the 7-vertex path. Relaxing tractability to the existence of a quasipolynomial-time algorithm, we know substantially more. Indeed, quasipolynomial-time algorithms were recently obtained for the $t$-vertex cycle, $C_t$ [Gartland et al., STOC '21] and the disjoint union of $t$ triangles, $tC_3$ [Bonamy et al., SODA '23]. We give, for every integer $t$, a polynomial-time algorithm running in $n^{O(t^5)}$ when $H$ is the friendship graph $K_1 + tK_2$ ($t$ disjoint edges plus a vertex fully adjacent to them), and a quasipolynomial-time algorithm running in $n^{O(t^2 \log n)+t^{O(1)}}$ when $H$ is $tC_3 \uplus C_4$ (the disjoint union of $t$ triangles and a 4-vertex cycle). The former extends a classical result on graphs excluding $tK_2$ as an induced subgraph [Alekseev, DAM '07], while the latter extends Bonamy et al.'s result.Comment: 15 pages, 2 figure

A note on connected greedy edge colouring

Following a given ordering of the edges of a graph $G$, the greedy edge colouring procedure assigns to each edge the smallest available colour. The minimum number of colours thus involved is the chromatic index $\chi'(G)$, and the maximum is the so-called Grundy chromatic index. Here, we are interested in the restricted case where the ordering of the edges builds the graph in a connected fashion. Let $\chi_c'(G)$ be the minimum number of colours involved following such an ordering. We show that it is NP-hard to determine whether $\chi_c'(G)>\chi'(G)$. We prove that $\chi'(G)=\chi_c'(G)$ if $G$ is bipartite, and that $\chi_c'(G)\leq 4$ if $G$ is subcubic.Comment: Comments welcome, 12 page

A Tight Local Algorithm for the Minimum Dominating Set Problem in Outerplanar Graphs

We show that there is a deterministic local algorithm (constant-time distributed graph algorithm) that finds a 5-approximation of a minimum dominating set on outerplanar graphs. We show there is no such algorithm that finds a (5-ε)-approximation, for any ε > 0. Our algorithm only requires knowledge of the degree of a vertex and of its neighbors, so that large messages and unique identifiers are not needed

Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

A graph is $O_k$-free if it does not contain $k$ pairwise vertex-disjoint and non-adjacent cycles. We show that Maximum Independent Set and 3-Coloring in $O_k$-free graphs can be solved in quasi-polynomial time. As a main technical result, we establish that "sparse" (here, not containing large complete bipartite graphs as subgraphs) $O_k$-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is proven sharp as there is an infinite family of $O_2$-free graphs without $K_{3,3}$-subgraph and whose treewidth is (at least) logarithmic. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in sparse $O_k$-free graphs, and that deciding the $O_k$-freeness of sparse graphs is polynomial time solvable.Comment: 28 pages, 6 figures. v3: improved complexity result