arXiv subjects

Bogdan Alecu

Publications and source records attributed to Bogdan Alecu.

21 records · Page 2Linked to original sources

arXiv2020

Critical properties of bipartite permutation graphs

The class of bipartite permutation graphs enjoys many nice and important properties. In particular, this class is critically important in the study of clique- and rank-width of graphs, because it is one of the minimal hereditary classes of graphs of unbounded clique- and rank-width. It also contains a number of important subclasses, which are critical with respect to other parameters, such as graph lettericity or shrub-depth, and with respect to other notions, such as well-quasi-ordering or complexity of algorithmic problems. In the present paper we identify critical subclasses of bipartite permutation graphs of various types.

math.CO↗

arXiv2018

Graph functionality

Let $G=(V,E)$ be a graph and $A$ its adjacency matrix. We say that a vertex $y \in V$ is a function of vertices $x_1, \ldots, x_k \in V$ if there exists a Boolean function $f$ of $k$ variables such that for any vertex $z \in V - \{y, x_1, \ldots, x_k\}$, $A(y,z)=f(A(x_1,z),\ldots,A(x_k,z))$. The functionality $fun(y)$ of vertex $y$ is the minimum $k$ such that $y$ is a function of $k$ vertices. The functionality $fun(G)$ of the graph $G$ is $\max\limits_H\min\limits_{y\in V(H)}fun(y)$, where the maximum is taken over all induced subgraphs $H$ of $G$. In the present paper, we show that functionality generalizes simultaneously several other graph parameters, such as degeneracy or clique-width, by proving that bounded degeneracy or bounded clique-width imply bounded functionality. Moreover, we show that this generalization is proper by revealing classes of graphs of unbounded degeneracy and clique-width, where functionality is bounded by a constant. This includes permutation graphs, unit interval graphs and line graphs. We also observe that bounded functionality implies bounded VC-dimension, i.e. graphs of bounded VC-dimension extend graphs of bounded functionality, and this extension is also proper.

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arXiv2018

Letter graphs and geometric grid classes of permutations: characterization and recognition

In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a non-constructive way, a polynomial-time recognition of geometric grid classes of permutations and $k$-letter graphs for a fixed $k$. However, constructive algorithms are available only for $k=2$. In this paper, we present the first constructive polynomial-time algorithm for the recognition of $3$-letter graphs. It is based on a structural characterization of graphs in this class.

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