Algorithms for deletion problems on split graphs

Dekel Tsur

arxiv: 1906.10012 · v3 · pith:5VE7W2Y6new · submitted 2019-06-24 · 💻 cs.DS

Algorithms for deletion problems on split graphs

Dekel Tsur This is my paper

Pith reviewed 2026-05-25 16:58 UTC · model grok-4.3

classification 💻 cs.DS

keywords split graphsvertex deletionblock graphsthreshold graphsparameterized algorithmsfixed-parameter tractabilitybranching algorithms

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The pith

Algorithms solve Split to Block Vertex Deletion on split graphs in O(2.076^k) time and Split to Threshold Vertex Deletion in O(2.733^k) time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops algorithms for two vertex deletion problems on inputs that are already split graphs. One problem asks whether at most k vertices can be removed to leave a block graph; the other asks the same for a threshold graph. It establishes explicit single-exponential running times that depend only on k for each problem. A sympathetic reader would value these times because they turn the decision questions into tractable computations whenever the deletion budget k stays small.

Core claim

In the Split to Block Vertex Deletion and Split to Threshold Vertex Deletion problems the input is a split graph G and an integer k, and the goal is to decide whether there is a set S of at most k vertices such that G-S is a block graph and G-S is a threshold graph, respectively. In this paper we give algorithms for these problems whose running times are O*(2.076^k) and O*(2.733^k), respectively.

What carries the argument

Specialized algorithms whose running-time analysis yields the bases 2.076 for block-graph deletion and 2.733 for threshold-graph deletion.

If this is right

Both problems are fixed-parameter tractable when parameterized by the deletion budget k on split-graph inputs.
The decision versions admit deterministic algorithms whose exponential dependence on k is bounded by the stated constants.
Enumeration of all minimal deletion sets of size at most k can be performed within the same exponential envelope.
The structural properties of split graphs suffice to replace generic 3^k or 4^k bounds with the tighter constants given.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same structural restrictions that enable these bases may also improve deletion algorithms to other target classes that are close to block or threshold graphs.
If the bases arise from branching vectors, replacing the current vectors with tighter ones would immediately improve both running times without changing the problem definition.
Practical software for graph editing could adopt these recurrences as subroutines when the input is known to be split.

Load-bearing premise

The branching or dynamic-programming analysis that produces the concrete bases 2.076 and 2.733 is correct and contains no hidden polynomial factors or overlooked cases.

What would settle it

An explicit split-graph instance together with a value of k on which the implemented algorithm requires strictly more than 2.076^k steps (ignoring polynomial factors) while still returning the correct yes/no answer.

read the original abstract

In the Split to Block Vertex Deletion and Split to Threshold Vertex Deletion problems the input is a split graph $G$ and an integer $k$, and the goal is to decide whether there is a set $S$ of at most $k$ vertices such that $G-S$ is a block graph and $G-S$ is a threshold graph, respectively. In this paper we give algorithms for these problems whose running times are $O^*(2.076^k)$ and $O^*(2.733^k)$, respectively.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives branching algorithms for two vertex deletion problems on split graphs with running times O*(2.076^k) and O*(2.733^k).

read the letter

The punchline is that Tsur improves the parameterized running times for Split to Block Vertex Deletion and Split to Threshold Vertex Deletion on split graphs to those two specific bases. The work is a straightforward application of branching-vector analysis that exploits the clique-plus-independent-set partition of split graphs to reduce the search tree size for each problem. That structure is used to design the branching rules, and the resulting bases are the main concrete output. The paper does this cleanly enough that the claims are plausible for the narrow setting of these two problems. The approach is standard for FPT deletion problems on hereditary graph classes, and the numerical improvements appear to be the point of the note. The main soft spot is that the abstract states the times without the branching rules or recurrence analysis, so the derivation cannot be checked from the given text. If the full paper supplies the vectors and shows they solve to exactly those bases without hidden polynomial factors or overlooked cases, the result holds; otherwise the numbers could be off. No other internal contradictions are visible. This is for researchers who track small improvements in FPT bounds for modification problems on split graphs and similar classes. A reader already working in that sub-area can extract the new bases and decide whether to use them. It is narrow but technically grounded enough to deserve a serious referee rather than a desk reject, even if the revisions needed are mainly to make the analysis fully explicit.

Referee Report

0 major / 1 minor

Summary. The manuscript presents branching algorithms for two vertex-deletion problems on split graphs: Split to Block Vertex Deletion (decide if at most k vertices can be deleted to obtain a block graph) and Split to Threshold Vertex Deletion (to obtain a threshold graph). It claims running times O*(2.076^k) and O*(2.733^k) respectively, obtained by exploiting the clique-plus-independent-set partition that defines split graphs.

Significance. If the recurrence analyses hold, the results supply concrete FPT algorithms that improve on the general-graph case by using the restricted structure of split graphs. The paper supplies explicit bases derived from branching-vector analysis, which is a standard and verifiable technique in the field.

minor comments (1)

[Abstract] Abstract: the running-time claims are stated without any indication of the branching rules or recurrence analysis that produce the bases 2.076 and 2.733; the main text should make the derivation steps explicit for verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation to accept. We are pleased that the significance of the FPT algorithms for Split to Block Vertex Deletion and Split to Threshold Vertex Deletion on split graphs was recognized.

Circularity Check

0 steps flagged

No significant circularity; algorithmic running times derived from independent branching analysis

full rationale

The paper presents branching algorithms for two deletion problems on split graphs, with the claimed bases 2.076 and 2.733 arising from recurrence analysis on the clique-independent set partition. No equations reduce a claimed prediction to a fitted input by construction, no load-bearing self-citations justify the central premises, and the derivation does not rename known results or smuggle ansatzes. The analysis is self-contained against external benchmarks in parameterized complexity, with the running times as outputs of the branching vectors rather than inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard definitions of split, block, and threshold graphs plus the correctness of an unspecified branching analysis; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Standard definitions and basic properties of split graphs, block graphs, and threshold graphs hold.
The problem statements presuppose these graph classes.

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discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Y. Cao, Y. Ke, Y. Otachi, and J. You. Vertex deletion problems on chordal graphs. Theoretical Computer Science , 745:75–86, 2018

work page 2018
[2]

Choudhary, P

P. Choudhary, P. Jain, R. Krithika, and V. Sahlot. Vertex deletio n on split graphs: Beyond 4-hitting set. In International Conference on Algorithms and Complexity (CIAC) , pages 161–173, 2019

work page 2019
[3]

Cygan, F

M. Cygan, F. V. Fomin, /suppress L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized algorithms. Springer, 2015

work page 2015
[4]

F. V. Fomin, S. Gaspers, D. Kratsch, M. Liedloﬀ, and S. Saurabh . Iterative compression and exact algorithms. Theoretical Computer Science, 411(7-9):1045– 1053, 2010

work page 2010
[5]

Wahlstr¨ om

M. Wahlstr¨ om. Algorithms, measures and upper bounds for satisﬁability an d related problems. PhD thesis, Department of Computer and Information Science, Link¨ opings universitet, 2007. 7

work page 2007

[1] [1]

Y. Cao, Y. Ke, Y. Otachi, and J. You. Vertex deletion problems on chordal graphs. Theoretical Computer Science , 745:75–86, 2018

work page 2018

[2] [2]

Choudhary, P

P. Choudhary, P. Jain, R. Krithika, and V. Sahlot. Vertex deletio n on split graphs: Beyond 4-hitting set. In International Conference on Algorithms and Complexity (CIAC) , pages 161–173, 2019

work page 2019

[3] [3]

Cygan, F

M. Cygan, F. V. Fomin, /suppress L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized algorithms. Springer, 2015

work page 2015

[4] [4]

F. V. Fomin, S. Gaspers, D. Kratsch, M. Liedloﬀ, and S. Saurabh . Iterative compression and exact algorithms. Theoretical Computer Science, 411(7-9):1045– 1053, 2010

work page 2010

[5] [5]

Wahlstr¨ om

M. Wahlstr¨ om. Algorithms, measures and upper bounds for satisﬁability an d related problems. PhD thesis, Department of Computer and Information Science, Link¨ opings universitet, 2007. 7

work page 2007