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We present families of combinatorial classes described as trees with nodes that can carry one of two types of "flowers": integer partitions or integer compositions. Two parameters on the flowers of trees will be considered: the number of…

A trivalent diagram is a connected, two-colored bipartite graph (parallel edges allowed but not loops) such that every black vertex is of degree 1 or 3 and every white vertex is of degree 1 or 2, with a cyclic order imposed on every set of…

I present a simple algorithm for enumerating the trees generated by a Context Free Grammar (CFG). The algorithm uses a pairing function to form a bijection between CFG derivations and natural numbers, so that trees can be uniquely decoded…

The generation of trees with a specified tree edit distance has significant applications across various fields, including computational biology, structured data analysis, and image processing. Recently, generative networks have been…

Large tree structures are ubiquitous and real-world relational datasets often have information associated with nodes (e.g., labels or other attributes) and edges (e.g., weights or distances) that need to be communicated to the viewers. Yet,…

This article presents two novel algorithms for generating random increasing trees. The first algorithm efficiently generates strictly increasing binary trees using an ad hoc method. The second algorithm improves the recursive method for…

We construct tree-decompositions of graphs that distinguish all their k-blocks and tangles of order k, for any fixed integer k. We describe a family of algorithms to construct such decompositions, seeking to maximize their diversity subject…

In this article we describe an algorithm that can be applied for the generation of various classes of maps on orientable surfaces. It uses existing generators for abstract graphs and combines them with an efficient embedding and isomorphism…

We introduce a generalization of the well known graph (vertex) coloring problem, which we call the problem of \emph{component coloring of graphs}. Given a graph, the problem is to color the vertices using minimum number of colors so that…

A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph…

Tree-shaped graphical models are widely used for their tractability. However, they unfortunately lack expressive power as they require committing to a particular sparse dependency structure. We propose a novel class of generative models…

Pattern counting in graphs is a fundamental primitive for many network analysis tasks, and a number of methods have been developed for scaling subgraph counting to large graphs. Many real-world networks carry a natural notion of strength of…

Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfying approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that…

In this paper, we relate the seemingly unrelated concepts of treewidth and boxicity. Our main result is that, for any graph G, boxicity(G) <= treewidth(G) + 2. We also show that this upper bound is (almost) tight. Our result leads to…

Let $T$ be a weighted tree. The weight of a subtree $T_1$ of $T$ is defined as the product of weights of vertices and edges of $T_1$. We obtain a linear-time algorithm to count the sum of weights of subtrees of $T$. As applications, we…

Real-world networks process structured connections since they have non-trivial vertex degree correlation and clustering. Here we propose a toy model of structure formation in real-world weighted network. In our model, a network evolves by…

General graphs are difficult for learning due to their irregular structures. Existing works employ message passing along graph edges to extract local patterns using customized graph kernels, but few of them are effective for the integration…

We define a growing model of random graphs. Given a sequence of nonnegative integers $\{d_n\}_{n=0}^\infty$ with the property that $d_i\leq i$, we construct a random graph on countably infinitely many vertices $v_0,v_1\ldots$ by the…

The discovery and analysis of network patterns are central to the scientific enterprise. In the present work, we developed and evaluated a new approach that learns the building blocks of graphs that can be used to understand and generate…

Motivated by the question of how macromolecules assemble, the notion of an {\it assembly tree} of a graph is introduced. Given a graph $G$, the paper is concerned with enumerating the number of assembly trees of $G$, a problem that applies…