Linear-Delay Enumeration for Minimal Steiner Problems

Kimelfeld and Sagiv [Kimelfeld and Sagiv, PODS 2006], [Kimelfeld and Sagiv, Inf. Syst. 2008] pointed out the problem of enumerating $K$-fragments is of great importance in a keyword search on data graphs. In a graph-theoretic term, the problem corresponds to enumerating minimal Steiner trees in (directed) graphs. In this paper, we propose a linear-delay and polynomial-space algorithm for enumerating all minimal Steiner trees, improving on a previous result in [Kimelfeld and Sagiv, Inf. Syst. 2008]. Our enumeration algorithm can be extended to other Steiner problems, such as minimal Steiner forests, minimal terminal Steiner trees, and minimal directed Steiner trees. As another variant of the minimal Steiner tree enumeration problem, we study the problem of enumerating minimal induced Steiner subgraphs. We propose a polynomial-delay and exponential-space enumeration algorithm of minimal induced Steiner subgraphs on claw-free graphs. Contrary to these tractable results, we show that the problem of enumerating minimal group Steiner trees is at least as hard as the minimal transversal enumeration problem on hypergraphs

Efficient Enumeration of Induced Subtrees in a K-Degenerate Graph

In this paper, we address the problem of enumerating all induced subtrees in an input k-degenerate graph, where an induced subtree is an acyclic and connected induced subgraph. A graph G = (V, E) is a k-degenerate graph if for any its induced subgraph has a vertex whose degree is less than or equal to k, and many real-world graphs have small degeneracies, or very close to small degeneracies. Although, the studies are on subgraphs enumeration, such as trees, paths, and matchings, but the problem addresses the subgraph enumeration, such as enumeration of subgraphs that are trees. Their induced subgraph versions have not been studied well. One of few example is for chordless paths and cycles. Our motivation is to reduce the time complexity close to O(1) for each solution. This type of optimal algorithms are proposed many subgraph classes such as trees, and spanning trees. Induced subtrees are fundamental object thus it should be studied deeply and there possibly exist some efficient algorithms. Our algorithm utilizes nice properties of k-degeneracy to state an effective amortized analysis. As a result, the time complexity is reduced to O(k) time per induced subtree. The problem is solved in constant time for each in planar graphs, as a corollary

Classical and consecutive pattern avoidance in rooted forests

Following Anders and Archer, we say that an unordered rooted labeled forest avoids the pattern $\sigma\in\mathcal{S}_k$ if in each tree, each sequence of labels along the shortest path from the root to a vertex does not contain a subsequence with the same relative order as $\sigma$. For each permutation $\sigma\in\mathcal{S}_{k-2}$, we construct a bijection between $n$-vertex forests avoiding $(\sigma)(k-1)k=\sigma(1)\cdots\sigma(k-2)(k-1)k$ and $n$-vertex forests avoiding $(\sigma)k(k-1)=\sigma(1)\cdots\sigma(k-2)k(k-1)$, giving a common generalization of results of West on permutations and Anders--Archer on forests. We further define a new object, the forest-Young diagram, which we use to extend the notion of shape-Wilf equivalence to forests. In particular, this allows us to generalize the above result to a bijection between forests avoiding $\{(\sigma_1)k(k-1), (\sigma_2)k(k-1), \dots, (\sigma_\ell) k(k-1)\}$ and forests avoiding $\{(\sigma_1)(k-1)k, (\sigma_2)(k-1)k, \dots, (\sigma_\ell) (k-1)k\}$ for $\sigma_1, \dots, \sigma_\ell \in \mathcal{S}_{k-2}$. Furthermore, we give recurrences enumerating the forests avoiding $\{123\cdots k\}$, $\{213\}$, and other sets of patterns. Finally, we extend the Goulden--Jackson cluster method to study consecutive pattern avoidance in rooted trees as defined by Anders and Archer. Using the generalized cluster method, we prove that if two length-$k$ patterns are strong-c-forest-Wilf equivalent, then up to complementation, the two patterns must start with the same number. We also prove the surprising result that the patterns $1324$ and $1423$ are strong-c-forest-Wilf equivalent, even though they are not c-Wilf equivalent with respect to permutations.Comment: 39 pages, 11 figure

Balanced binary trees in the Tamari lattice

We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T0, T1] where T0 and T1 are balanced trees are isomorphic as posets to a hypercube. We introduce tree patterns and synchronous grammars to get a functional equation of the generating series enumerating balanced tree intervals

Pattern avoidance in binary trees

This paper considers the enumeration of trees avoiding a contiguous pattern. We provide an algorithm for computing the generating function that counts n-leaf binary trees avoiding a given binary tree pattern t. Equipped with this counting mechanism, we study the analogue of Wilf equivalence in which two tree patterns are equivalent if the respective n-leaf trees that avoid them are equinumerous. We investigate the equivalence classes combinatorially. Toward establishing bijective proofs of tree pattern equivalence, we develop a general method of restructuring trees that conjecturally succeeds to produce an explicit bijection for each pair of equivalent tree patterns.Comment: 19 pages, many images; published versio