| Summary: | A graph $G$ is signed if each edge is assigned $+$ or $-$. A signed graph is
balanced if there is a bipartition of its vertex set such that an edge has sign
$-$ if and only if its endpoints are in different parts. The Edwards-Erd\"os
bound states that every graph with $n$ vertices and $m$ edges has a balanced
subgraph with at least $\frac{m}{2}+\frac{n-1}{4}$ edges. In the Signed Max Cut
Above Tight Lower Bound (Signed Max Cut ATLB) problem, given a signed graph $G$
and a parameter $k$, the question is whether $G$ has a balanced subgraph with
at least $\frac{m}{2}+\frac{n-1}{4}+\frac{k}{4}$ edges. This problem
generalizes Max Cut Above Tight Lower Bound, for which a kernel with $O(k^5)$
vertices was given by Crowston et al. [ICALP 2012, Algorithmica 2015]. Crowston
et al. [TCS 2013] improved this result by providing a kernel with $O(k^3)$
vertices for the more general Signed Max Cut ATLB problem. In this article we
are interested in improving the size of the kernels for Signed Max Cut ATLB on
restricted graph classes for which the problem remains hard. For two integers
$r,\ell \geq 0$, a graph $G$ is an $(r,\ell)$-graph if $V(G)$ can be
partitioned into $r$ independent sets and $\ell$ cliques. Building on the
techniques of Crowston et al. [TCS 2013], we provide a kernel with $O(k^2)$
vertices on $(r,\ell)$-graphs for any fixed $r,\ell \geq 0$, and a simple
linear kernel on subclasses of split graphs for which we prove that the problem
is still NP-hard.
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