Polynomial kernels for edge modification problems towards block and strictly chordal graphs

TL;DR

The paper addresses edge-editing problems that transform a given graph into a block graph or a strictly chordal graph, showing NP-hardness for most variants while delivering polynomial kernels with concrete bounds: $O(k^2)$ for Block Graph Editing/Deletion, $O(k^3)$ for Strictly Chordal Completion/Deletion, and $O(k^4)$ for Strictly Chordal Editing. It develops a framework built on join compositions, the critical clique graph, and structured BG- and SC-branches to bound the number of affected components and the total graph size after reductions. The main technical contributions are the kernelization algorithms for Block Graph Editing/Deletion and Strictly Chordal Editing, plus refined kernels for the completion and deletion variants, all proven safe and computable in polynomial time. These results advance the understanding of kernelization for edge modification problems in chordal graph families and hint at broader generic approaches for such problems.

Abstract

We consider edge modification problems towards block and strictly chordal graphs, where one is given an undirected graph $G = (V,E)$ and an integer $k \in \mathbb{N}$ and seeks to edit (add or delete) at most $k$ edges from $G$ to obtain a block graph or a strictly chordal graph. The completion and deletion variants of these problems are defined similarly by only allowing edge additions for the former and only edge deletions for the latter. Block graphs are a well-studied class of graphs and admit several characterizations, e.g. they are diamond-free chordal graphs. Strictly chordal graphs, also referred to as block duplicate graphs, are a natural generalization of block graphs where one can add true twins of cut-vertices. Strictly chordal graphs are exactly dart and gem-free chordal graphs. We prove the NP-completeness for most variants of these problems and provide $O(k^2)$ vertex-kernels for Block Graph Editing and Block Graph Deletion, $O(k^3)$ vertex-kernels for Strictly Chordal Completion and Strictly Chordal Deletion and a $O(k^4)$ vertex-kernel for Strictly Chordal Editing.

Figures (9)

• Figure 1: Kernelization status of subclasses of chordal graphs. A class below the red or blue line indicates that the corresponding completion, deletion or editing problem admits a polynomial kernel. All the edge modification problems for the presented classes are NP-complete at the exception of the ones for $4$-leaf power for which the complexity is unknown, Split Editing which is surprisingly in P hammer1981splittance, and Block Graph Completion which is also in $P$ as observed previously.
• Figure 2: The diamond, dart, gem and hole (cycle of length at least $4$)
• Figure 3: Example of a graph $G= (G_1, S_1) \otimes (\bigcup_{2 \leqslant i \leqslant 4}G_i, \bigcup_{2 \leqslant i \leqslant 4}S_i)$ as constructed in \ref{['obs:SCmulticomp']}. There, $S_1$ intersects exactly one maximal clique of $G_1$ and is a critical clique of $G$; $S_2$ intersects exactly one maximal clique of $G_2$; $S_3$ is a maximal clique of $G_3$; $S_4$ is a critical clique of $G_4$.
• Figure 4: Illustration of the case where $p_1$ and $p_2$ are not in the same connected component of $H'$ in the \ref{['lem:2BGbranch']}. The white circles correspond to critical cliques of $B$. The blue sets correspond to $N_B(P_1)$ and $N_B(P_2)$.
• Figure 5: Illustration of the case where $p_1$ and $p_2$ are in the same connected component of $H'$ and $F$ contains a min-cut of $B$ in \ref{['lem:2-SC-branchE']}. The white circles correspond to critical cliques of $B$. The blue sets correspond to $N_B(P_1)$ and $N_B(P_2)$. The red dotted edges between $B_1$ and $B_2$ are edges of an edge-cut of $B$ deleted by the edition.

Theorems & Definitions (60)

• Conjecture 1: BP13
• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1
• Definition 2: Critical clique graph
• Theorem 4: Theorem 8 MW15
• Lemma 1: BPP10
• Definition 3
• Lemma 2