Graph modification of bounded size to minor-closed classes as fast as vertex deletion

TL;DR

To the best of the knowledge, these are the first parameterized algorithms with a reasonable parametric dependence for such a general family of graph modification problems to minor-closed classes.

Abstract

A replacement action is a function $\mathcal{L}$ that maps each graph $H$ to a collection of graphs of size at most $|V(H)|$. Given a graph class $\mathcal{H}$, we consider a general family of graph modification problems, called $\mathcal{L}$-Replacement to $\mathcal{H}$, where the input is a graph $G$ and the question is whether it is possible to replace some induced subgraph $H_1$ of $G$ on at most $k$ vertices by a graph $H_2$ in $\mathcal{L}(H_1)$ so that the resulting graph belongs to $\mathcal{H}$. $\mathcal{L}$-Replacement to $\mathcal{H}$ can simulate many graph modification problems including vertex deletion, edge deletion/addition/edition/contraction, vertex identification, subgraph complementation, independent set deletion, (induced) matching deletion/contraction, etc. We present two algorithms. The first one solves $\mathcal{L}$-Replacement to $\mathcal{H}$ in time $2^{{\rm poly}(k)}\cdot |V(G)|^2$ for every minor-closed graph class $\mathcal{H}$, where {\rm poly} is a polynomial whose degree depends on $\mathcal{H}$, under a mild technical condition on $\mathcal{L}$. This generalizes the results of Morelle, Sau, Stamoulis, and Thilikos [ICALP 2020, ICALP 2023] for the particular case of Vertex Deletion to $\mathcal{H}$ within the same running time. Our second algorithm is an improvement of the first one when $\mathcal{H}$ is the class of graphs embeddable in a surface of Euler genus at most $g$ and runs in time $2^{\mathcal{O}(k^{9})}\cdot |V(G)|^2$, where the $\mathcal{O}(\cdot)$ notation depends on $g$. To the best of our knowledge, these are the first parameterized algorithms with a reasonable parametric dependence for such a general family of graph modification problems to minor-closed classes.

Figures (16)

• Figure 1: Example of $(H_2,\phi)\in\mathcal{L}(H_1)$ and of the graph modification $G_{(H,\phi)}^S$ where $S$ is the set of black vertices of $G$. $\phi$ is represented by the colors, that is, $\phi(u_1)=\phi(u_5)=v_1$, $\phi(u_2)=\phi(v_2)$, $\phi(u_3)=\emptyset$, and $\phi(u_4)=v_3$. The order on the vertex sets of the depicted graphs is given by the corresponding labels.
• Figure 2: If $\mathcal{L}$ is hereditary, then a restriction of an allowed modification is also allowed.
• Figure 3: A $5$-wall. Its first layer is depicted in red and its second layer in orange. Its central vertices are depicted in a green square.
• Figure 4: A rendition. The cells are depicted in light orange.
• Figure 5: Illustration of a flatness pair $(W,\frak{R})$ of a graph $G$ (adapted from BasteST23hittIV). The edges of $W$ are depicted in orange and the $\mathcal{R}$-compass of $W$ is the union of all parts of $G$ that are drawn in light orange cells. The yellow vertices are the vertices of $V(\Omega)$ and the squared vertices are the choice of pegs (in purple) and corners (in black) of $W$.

Theorems & Definitions (46)

• proof
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Proposition 3.1: SauST22kapiII
• Proposition 3.2: KawarabayashiTW18anew
• Proposition 3.3: SauST24amor
• Proposition 3.4: SauST24amor
• Proposition 3.5: SauST24amor