arXiv subjects

Toni Bohnlein

Publications and source records attributed to Toni Bohnlein.

3 recordsLinked to original sources

arXiv2025

Degree Realization by Bipartite Cactus Graphs

The \textsc{Degree Realization} problem with respect to a graph family $\mathcal{F}$ is defined as follows. The input is a sequence $d$ of $n$ positive integers, and the goal is to decide whether there exists a graph $G \in \mathcal{F}$ whose degrees correspond to $d$. The main challenges are to provide a precise characterization of all the sequences that admit a realization in $\mathcal{F}$ and to design efficient algorithms that construct one of the possible realizations, if one exists. This paper studies the problem of realizing degree sequences by bipartite cactus graphs (where the input is given as a single sequence, without the bi-partition). A characterization of the sequences that have a cactus realization is already known [28]. In this paper, we provide a systematic way to obtain such a characterization, accompanied by a realization algorithm. This allows us to derive a characterization for bipartite cactus graphs, and as a byproduct, also for several other interesting sub-families of cactus graphs, including bridge-less cactus graphs and core cactus graphs, as well as for the bipartite sub-families of these families.

cs.DM↗

arXiv2025

Degree Realization by Bipartite Multigraphs

The problem of realizing a given degree sequence by a multigraph can be thought of as a relaxation of the classical degree realization problem (where the realizing graph is simple). This paper concerns the case where the realizing multigraph is required to be bipartite. The problem of characterizing sequences that can be realized by a bipartite graph has two variants. In the simpler one, termed BDR$^P$, the partition of the sequence into two sides is given as part of the input. A complete characterization for realizability in this variant was given by Gale and Ryser over sixty years ago. However, the variant where the partition is not given, termed BDR, is still open. For bipartite multigraph realizations, there are also two variants. For BDR$^P$, where the partition is given as part of the input, a characterization was known for determining whether there is a multigraph realization whose underlying graph is bipartite, such that the maximum number of copies of an edge is at most $r$. We present a characterization for determining if there is a bipartite multigraph realization such that the total number of excess edges is at most $t$. We show that optimizing these two measures may lead to different realizations, and that optimizing by one measure may increase the other substantially. As for the variant BDR, where the partition is not given, we show that determining whether a given (single) sequence admits a bipartite multigraph realization is NP-hard. Moreover, we show that this hardness result extends to any graph family which is a sub-family of bipartite graphs and a super-family of paths. On the positive side, we provide an algorithm that computes optimal realizations for the case where the number of balanced partitions is polynomial, and present sufficient conditions for the existence of bipartite multigraph realizations that depend only on the largest degree of the sequence.

math.CO↗

arXiv2024

Approximate Realizations for Outerplanaric Degree Sequences

We study the question of whether a sequence d = (d_1,d_2, \ldots, d_n) of positive integers is the degree sequence of some outerplanar (a.k.a. 1-page book embeddable) graph G. If so, G is an outerplanar realization of d and d is an outerplanaric sequence. The case where \sum d \leq 2n - 2 is easy, as d has a realization by a forest (which is trivially an outerplanar graph). In this paper, we consider the family \cD of all sequences d of even sum 2n\leq \sum d \le 4n-6-2\multipl_1, where \multipl_x is the number of x's in d. (The second inequality is a necessary condition for a sequence d with \sum d\geq 2n to be outerplanaric.) We partition \cD into two disjoint subfamilies, \cD=\cD_{NOP}\cup\cD_{2PBE}, such that every sequence in \cD_{NOP} is provably non-outerplanaric, and every sequence in \cD_{2PBE} is given a realizing graph $G$ enjoying a 2-page book embedding (and moreover, one of the pages is also bipartite).

cs.DS↗