Algorithmic and structural aspects of graph parameters

Abstract

For the algorithmic aspects of graph parameters, we first present a polynomial-time algorithm approximating the minimum number of vertices of an input graph whose removal leads to a ptolemaic graph within a constant factor. For a family $\mathcal{F}$ of graphs and an integer $r$, the $(r,\mathcal{F})$-covering number of a graph $G$ is the minimum size of a set $D\subseteq V(G)$ such that every induced subgraph of $G$ isomorphic to a graph in $\mathcal{F}$ is at distance at most $r$ from $D$, and the $(r,\mathcal{F})$-packing number of $G$ is the maximum size of a collection of subsets $A_1,\ldots A_\ell$ of $V(G)$ such that each of them induces a subgraph of $G$ isomorphic to a graph in $\mathcal{F}$, and for all $1\leq i<j\leq \ell$, the distance in $G$ between $A_i$ and $A_j$ is larger than $r$. For every finite family $\mathcal{F}$ of connected graphs, we present polynomial-time algorithms converting an input graph $G$ to a smaller graph $G'$ such that $G$ has $(r,\mathcal{F})$-covering or $(r,\mathcal{F})$-packing number at most $k$ for an integer $k$ if and only if $G'$ has $(r,\mathcal{F})$-covering or $(r,\mathcal{F})$-packing number at most $k+1$, respectively.

For the structural aspects of graph parameters, we present several bounds for the twin-width of graphs, which was introduced by Bonnet, Kim, Thomass\'{e}, and Watrigant (J. ACM, 2022). First, we present two upper bounds for the twin-width of a graph in terms of the number of vertices and the number of edges, respectively. We also show that the upper bound in terms of the number of vertices is asymptotically tight by computing the twin-width of Paley graphs. Second, we asymptotically determine the twin-width of random graphs and present threshold functions for small twin-width of random graphs. Last, for every integer $d\leq3$, we determine the full list of minimal forbidden minors of multigraphs $G$ such that every graph obtained from $G$ by replacing each edge by a path of length at least $3$ has twin-width at most $d$.