arXiv subjects

Christian Rosenke

Publications and source records attributed to Christian Rosenke.

6 recordsLinked to original sources

arXiv2026

Flood-It with Jewelry -- Characterizing the Game Complexity for Cograph Generalizations

Flood-It is a single-player game played on a precolored graph $G$, where the objective is to make $G$ monochromatic using as few flooding moves as possible. In each move, a color $c$ is selected and all vertices reachable from a fixed pivot vertex via a monochromatic path are recolored with $c$. In the free variant, the pivot may be chosen anew in every move. Deciding whether a graph can be made monochromatic in at most $k$ moves is NP-complete for both variants, fixed and free. This hardness persists even under strong structural restrictions such as split graphs and trees. The Free Flood-It variant is generally considered more difficult than its fixed-pivot counterpart, as it remains hard on several graph classes where the latter becomes tractable, including co-comparability and AT-free graphs. Cographs, that is, $P_4$-free graphs, are among the few classes on which even Free Flood-It is solvable in polynomial time and therefore serve as our starting point. We consider the ten natural one-vertex extensions of $P_4$ -- referred to as jewels -- and study the complexity of both flooding games on the $1024$ graph classes obtained by forbidding subsets of these graphs as induced subgraphs. Our main contribution is a polynomial-time algorithm for Free Flood-It on graphs that are free of the three jewels bull, gem, and $P_5$, covering $128$ of the $1024$ classes. In addition, we prove that both variants remain NP-complete on thin-spider graphs, which exclude the eight jewels banner, co-banner, chair, gem, house, kite, $P_5$, and $C_5$, thereby establishing hardness for $256$ additional classes. Combined with known algorithms and hardness results, our work determines the complexity of both Flood-It variants for $896$ of the $1024$ considered graph classes.

cs.DM↗

arXiv2026

Fanciful Figurines flip Free Flood-It -- Polynomial-Time Miniature Painting on Co-gem-free Graphs

Inspired by the eponymous hobby, we introduce Miniature Painting as the computational problem to paint a given graph $G=(V,E)$ according to a prescribed template $t \colon V \rightarrow C$, which assigns colors $C$ to the vertices of $G$. In this setting, the goal is to realize the template using a shortest possible sequence of brush strokes, where each stroke overwrites a connected vertex subset with a color in $C$. We show that this problem is equivalent to a reversal of the well-studied Free Flood-It game, in which a colored graph is decolored into a single color using as few moves as possible. This equivalence allows known complexity results for Free Flood-It to be transferred directly to Miniature Painting, including NP-hardness under severe structural restrictions, such as when $G$ is a grid, a tree, or a split graph. Our main contribution is a polynomial-time algorithm for Miniature Painting on graphs that are free of induced co-gems, a graph class that strictly generalizes cographs. As a direct consequence, Free Flood-It is also polynomial-time solvable on co-gem-free graphs, independent of the initial coloring.

cs.DS↗

arXiv2023

Computing Optimal Leaf Roots of Chordal Cographs in Linear Time

A graph G is a k-leaf power, for an integer k >= 2, if there is a tree T with leaf set V(G) such that, for all vertices x, y in V(G), the edge xy exists in G if and only if the distance between x and y in T is at most k. Such a tree T is called a k-leaf root of G. The computational problem of constructing a k-leaf root for a given graph G and an integer k, if any, is motivated by the challenge from computational biology to reconstruct phylogenetic trees. For fixed k, Lafond [SODA 2022] recently solved this problem in polynomial time. In this paper, we propose to study optimal leaf roots of graphs G, that is, the k-leaf roots of G with minimum k value. Thus, all k'-leaf roots of G satisfy k <= k'. In terms of computational biology, seeking optimal leaf roots is more justified as they yield more probable phylogenetic trees. Lafond's result does not imply polynomial-time computability of optimal leaf roots, because, even for optimal k-leaf roots, k may (exponentially) depend on the size of G. This paper presents a linear-time construction of optimal leaf roots for chordal cographs (also known as trivially perfect graphs). Additionally, it highlights the importance of the parity of the parameter k and provides a deeper insight into the differences between optimal k-leaf roots of even versus odd k. Keywords: k-leaf power, k-leaf root, optimal k-leaf root, trivially perfect leaf power, chordal cograph

math.CO↗

arXiv2018

Towards Completely Characterizing the Complexity of Boolean Nets Synthesis

Boolean nets are Petri nets that permit at most one token per place. Research has approached this important subject in many ways which resulted in various different classes of boolean nets. But yet, they are only distinguished by the allowed interactions between places and transitions, that is, the possible effects of firing transitions. There are eight different interactions: no operation (nop), input (inp), output (out), set, reset (res), swap, test of occupation (used), and test of disposability (free). Considering every combination for a possible net class yields 256 boolean classes in total. The synthesis problem for a particular class is to take an automaton A and compute a boolean net of that class that has a state graph isomorphic to A. To the best of our knowledge, the computational complexity of this problem has been analyzed for just two of the 256 classes, namely elementary nets systems (nop, inp, out), where the problem is NP-hard, and flip-flop nets (nop, inp, out, swap), which are synthesizable in polynomial time. However, depending on the desired net features, like read-only places, exception handling, or hierarchy, one has to synthesize nets with other interactions as for instance contextual nets (nop, inp, out, used, free) or trace nets (nop, inp, out, set, res, used, free). The contribution of this paper is a thorough investigation of the synthesis complexity for the 128 boolean net classes that allow nop. Our main result is a general proof scheme that identifies 77 NP-hard cases. All remaining 51 classes are shown to be synthesizable in polynomial time where 35 of them turn out to be trivial.

cs.CC↗

arXiv2017

The Hardness of Synthesizing Elementary Net Systems from Highly Restricted Inputs

Elementary net systems (ENS) are the most fundamental class of Petri nets. Their synthesis problem has important applications in the design of digital hardware and commercial processes. Given a labeled transition system (TS) $A$, feasibility is the NP-complete decision problem whether $A$ can be equivalently synthesized into an ENS. It is well known that $A$ is feasible if and only if it has the event state separation property (ESSP) and the state separation property (SSP). Recently, these properties have also been studied individually for their practical implications. A fast ESSP algorithm, for instance, would allow applications to at least validate the language equivalence of $A$ and a synthesized ENS. Being able to efficiently decide SSP, on the other hand, could serve as a quick-fail preprocessing mechanism for synthesis. Although a few tractable subclasses have been found, this paper destroys much of the hope that many practically meaningful input restrictions make feasibility or at least one of ESSP and SSP efficient. We show that all three problems remain NP-complete even if the input is restricted to linear TSs where every event occurs at most three times or if the input is restricted to TSs where each event occurs at most twice and each state has at most two successor and two predecessor states.

cs.LO↗

arXiv2014

Towards a Characterization of Leaf Powers by Clique Arrangements

The class ${\cal L}_k$ of $k$-leaf powers consists of graphs $G=(V,E)$ that have a $k$-leaf root, that is, a tree $T$ with leaf set $V$, where $xy \in E$, if and only if the $T$-distance between $x$ and $y$ is at most $k$. Structure and linear time recognition algorithms have been found for $2$-, $3$-, $4$-, and, to some extent, $5$-leaf powers, and it is known that the union of all $k$-leaf powers, that is, the graph class ${\cal L} = \bigcup_{k=2}^\infty {\cal L}_k$, forms a proper subclass of strongly chordal graphs. Despite from that, no essential progress has been made lately. In this paper, we use the new notion of clique arrangements to suggest that leaf powers are a natural special case of strongly chordal graphs. The clique arrangement ${\cal A}(G)$ of a chordal graph $G$ is a directed graph that represents the intersections between maximal cliques of $G$ by nodes and the mutual inclusion of these vertex subsets by arcs. Recently, strongly chordal graphs have been characterized as the graphs that have a clique arrangement without bad $k$-cycles for $k \geq 3$. We show that the clique arrangement of every graph of ${\cal L}$ is free of bad $2$-cycles. The question whether this characterizes the class ${\cal L}$ exactly remains open.

cs.DM↗