Let be a graph on vertices. For , a spanning forest of is called an ‐perfect forest if every tree... more Let be a graph on vertices. For , a spanning forest of is called an ‐perfect forest if every tree in is an induced subgraph of and exactly vertices of have even degree (including zero). An ‐perfect forest of is proper if it has no vertices of degree zero. Scott showed that every connected graph with an even number of vertices contains a (proper) 0‐perfect forest. We prove that one can find a 0‐perfect forest with a minimum number of edges in polynomial time, but it is NP‐hard to obtain a 0‐perfect forest with a maximum number of edges. Moreover, we show that to decide whether has a 0‐perfect forest with at least edges, where is the parameter, is W[1]‐hard. We also prove that for a prescribed edge of it is NP‐hard to obtain a 0‐perfect forest containing but one can decide if there exists a 0‐perfect forest not containing in polynomial time. It is easy to see that every connected graph with an odd number of vertices has a 1‐perfect forest. It is not the case for proper 1‐perfect fores...
From 8th to 13th July 2007, the Dagstuhl Seminar ``Structure Theory and FPT Algorithmics for Grap... more From 8th to 13th July 2007, the Dagstuhl Seminar ``Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
The famous Chinese Postman Problem (CPP) is polynomial time solvable on both undirected and direc... more The famous Chinese Postman Problem (CPP) is polynomial time solvable on both undirected and directed graphs. Gutin et al. [Discrete Applied Math 217 (2016)] generalized these results by proving that CPP on $c$-edge-colored graphs is polynomial time solvable for every $c\geq 2$. In CPP on weighted edge-colored graphs $G$, we wish to find a minimum weight properly colored closed walk containing all edges of $G$ (a walk is properly colored if every two consecutive edges are of different color, including the last and first edges in a closed walk). In this paper, we consider CPP on arc-colored digraphs (for properly colored closed directed walks), and provide a polynomial-time algorithm for the problem on weighted 2-arc-colored digraphs. This is a somewhat surprising result since it is NP-complete to decide whether a 2-arc-colored digraph has a properly colored directed cycle [Gutin et al., Discrete Math 191 (1998)]. To obtain the polynomial-time algorithm, we characterize 2-arc-colored ...
A weighted proper orientation of a given graph $G$, denoted by $(D,w)$, is an orientation $D$ wit... more A weighted proper orientation of a given graph $G$, denoted by $(D,w)$, is an orientation $D$ with a weight function $w: E(G)\rightarrow \mathbb{Z}_+$, such that the in-weight of any adjacent vertices are distinct, where the in-weight of $v$ in $D$, denoted by $w^-_D(v)$, is the sum of the weights of arcs towards $v$. The weighted proper orientation number of a graph $G$, denoted by $\overrightarrow{\chi}_w(G)$, is the minimum of maximum in-weight of $v$ in $D$ over all weighted proper orientation $(D,w)$ of $G$. This parameter was first introduced by Araujo et al. (2019). When the weights of all edges eqaul to one, this parameter is equal to the \textit{proper orientation number} of $G$. The optimal weighted proper orientation is a weighted proper orientation $(D,w)$ such that $\max_{v\in V(G)}w_D^-(v)=\overrightarrow{\chi}_w(G)$. Araujo et al. (2016) showed that $\overrightarrow{\chi}(G)\le 7$ for every cactus $G$ and the bound is tight. We prove that for every cactus $G$, $\overr...
In his paper “Kings in Bipartite Hypertournaments,” Petrovic stated two conjectures on 4‐kings in... more In his paper “Kings in Bipartite Hypertournaments,” Petrovic stated two conjectures on 4‐kings in multipartite hypertournaments. We prove one of these conjectures and give counterexamples for the other.
Let be a graph on vertices. For , a spanning forest of is called an ‐perfect forest if every tree... more Let be a graph on vertices. For , a spanning forest of is called an ‐perfect forest if every tree in is an induced subgraph of and exactly vertices of have even degree (including zero). An ‐perfect forest of is proper if it has no vertices of degree zero. Scott showed that every connected graph with an even number of vertices contains a (proper) 0‐perfect forest. We prove that one can find a 0‐perfect forest with a minimum number of edges in polynomial time, but it is NP‐hard to obtain a 0‐perfect forest with a maximum number of edges. Moreover, we show that to decide whether has a 0‐perfect forest with at least edges, where is the parameter, is W[1]‐hard. We also prove that for a prescribed edge of it is NP‐hard to obtain a 0‐perfect forest containing but one can decide if there exists a 0‐perfect forest not containing in polynomial time. It is easy to see that every connected graph with an odd number of vertices has a 1‐perfect forest. It is not the case for proper 1‐perfect fores...
From 8th to 13th July 2007, the Dagstuhl Seminar ``Structure Theory and FPT Algorithmics for Grap... more From 8th to 13th July 2007, the Dagstuhl Seminar ``Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
The famous Chinese Postman Problem (CPP) is polynomial time solvable on both undirected and direc... more The famous Chinese Postman Problem (CPP) is polynomial time solvable on both undirected and directed graphs. Gutin et al. [Discrete Applied Math 217 (2016)] generalized these results by proving that CPP on $c$-edge-colored graphs is polynomial time solvable for every $c\geq 2$. In CPP on weighted edge-colored graphs $G$, we wish to find a minimum weight properly colored closed walk containing all edges of $G$ (a walk is properly colored if every two consecutive edges are of different color, including the last and first edges in a closed walk). In this paper, we consider CPP on arc-colored digraphs (for properly colored closed directed walks), and provide a polynomial-time algorithm for the problem on weighted 2-arc-colored digraphs. This is a somewhat surprising result since it is NP-complete to decide whether a 2-arc-colored digraph has a properly colored directed cycle [Gutin et al., Discrete Math 191 (1998)]. To obtain the polynomial-time algorithm, we characterize 2-arc-colored ...
A weighted proper orientation of a given graph $G$, denoted by $(D,w)$, is an orientation $D$ wit... more A weighted proper orientation of a given graph $G$, denoted by $(D,w)$, is an orientation $D$ with a weight function $w: E(G)\rightarrow \mathbb{Z}_+$, such that the in-weight of any adjacent vertices are distinct, where the in-weight of $v$ in $D$, denoted by $w^-_D(v)$, is the sum of the weights of arcs towards $v$. The weighted proper orientation number of a graph $G$, denoted by $\overrightarrow{\chi}_w(G)$, is the minimum of maximum in-weight of $v$ in $D$ over all weighted proper orientation $(D,w)$ of $G$. This parameter was first introduced by Araujo et al. (2019). When the weights of all edges eqaul to one, this parameter is equal to the \textit{proper orientation number} of $G$. The optimal weighted proper orientation is a weighted proper orientation $(D,w)$ such that $\max_{v\in V(G)}w_D^-(v)=\overrightarrow{\chi}_w(G)$. Araujo et al. (2016) showed that $\overrightarrow{\chi}(G)\le 7$ for every cactus $G$ and the bound is tight. We prove that for every cactus $G$, $\overr...
In his paper “Kings in Bipartite Hypertournaments,” Petrovic stated two conjectures on 4‐kings in... more In his paper “Kings in Bipartite Hypertournaments,” Petrovic stated two conjectures on 4‐kings in multipartite hypertournaments. We prove one of these conjectures and give counterexamples for the other.
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