Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.37295

A Linear Kernel for Planar Semitotal Domination

Paloma T. de Lima·Lukas Retschmeier
View original source ↗Read source document ↗Export metadata ↓

Abstract

A set of vertices $S$ in a graph $G$ is a semitotal dominating set if $S$ is a dominating set and every vertex in $S$ has a witness, that is, another vertex in $S$ within distance at most two. The Semitotal Domination problem asks whether $G$ admits such a set of size at most $k$. We study this problem parameterized by the solution size. We provide a linear kernel of size $358k$ on planar graphs, a result that adds to several existing linear kernels for other domination variants on planar graphs. Additionally, we show Semitotal Domination is $W[2]$-hard on split graphs, on bipartite graphs and on $t$-claw-free graphs, it is $W[1]$-hard on circle graphs.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Paloma T. de Lima, Lukas Retschmeier. 2026-10-27. A Linear Kernel for Planar Semitotal Domination. https://arxiv.org/abs/2610.37295

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Interactive Particle Systems on Hypergraphs, Drift Analysis and the WalkSAT algorithm

We analyze the expected running time of WalkSAT, a well-known local search procedure for satisfiability solving, on satisfiable instances of the k-XOR SAT problem. We obtain estimates of this expected running time by reducing the problem to a setting amenable to classical techniques from drift analysis. A crucial ingredient of this reduction is the definition of (new, explosive) hypergraph versions of interacting particle systems, notably of coalescing and annihilating random walks as well as the voter model. The use of these tools allows to show that the expected running time of WalkSAT depends on structural parameter (we call odd Cheeger drift) of the dual of the formula hypergraph.

cs.DS↗
arXiv2026

A Better Analysis For PPSZ For 3-SAT

The PPSZ algorithm of Paturi, Pudlák, Saks, and Zane solves 3-SAT by processing the variables in a uniformly random order, setting a variable whenever a bounded number of clauses implies its value, and guessing it otherwise. Its running time is determined by how many variables are forced in this process, and the exact exponent is a central question in the study of exponential-time satisfiability algorithms. Scheder showed that the classical analysis of PPSZ is not tight for 3-SAT. We ask how much further a sharper analysis of the unmodified algorithm can go. Our analysis uses auxiliary distributions on variable orderings that correlate the positions of variables sharing critical clauses. Such correlations make forcing more likely for some variables, can make it less likely for others, and incur an entropy cost. We choose the correlated pairs at random on short components, so that every pair is selected with the same probability, and we express the forcing loss as a sum of weights attached to these pairs in advance. Gains and losses then scale by the same factor and can be compared directly. Combining the resulting estimate with the complementary estimate of Scheder through an explicit linear-programming certificate gives a uniform improvement for every uniquely satisfiable formula. We prove that the original PPSZ algorithm, with a sufficiently large but fixed implication strength, solves Unique-3-SAT in randomized time $O^*(1.306969598^n)$. Through the lifting framework of Scheder and Steinberger, it solves general 3-SAT in randomized time $O^*(1.307031578^n)$. Both bounds improve the guarantees obtained from the analysis of Scheder for the same algorithm. These improvements follow entirely from the analysis, while PPSZ continues to use a uniformly random variable order and unbiased guesses. It shows that the known guarantees for PPSZ can be strengthened by analysis alone.

cs.DS↗
arXiv2026

Online Service with Per-Batch Maximum Delays

We study online service with per-batch maximum delays on a finite metric space. Requests arrive over time, and a server serves each batch by following a walk visiting its requests' locations. Movement is instantaneous. The objective is to minimize the total distance traveled by the server plus the sum, over all batches, of the maximum waiting time in each batch. We give deterministic polynomial-time algorithms with competitive ratios $10$ on finite lines, $12$ on weighted trees, and $20$ on arbitrary finite metrics. Unlike additive delay, maximum delay allows younger requests to join a batch without increasing its delay charge. Our algorithms coordinate service using a common waiting clock and a threshold determined by the metric. The analysis uses certificates for groups of requests to compare this policy with any offline schedule, even one that serves different batches and follows different walks. This comparison shows that coordinating service in this way loses only a constant factor. The bounds hold whether the server must serve requests it visits or may leave them unserved; the two rules also have equal offline optimal costs. Offline optimization is polynomial on lines and explicitly represented weighted trees, and NP hard on general metrics. When arrivals are announced but locations are hidden until visited, we give an $84$-competitive algorithm on a known finite line under either rule.

cs.DS↗