Grzegorz Kubicki

An algorithm to find agreement subtrees

Given two binary trees, a largest subtree contained in both of the original trees that has been obtained by pruning vertices is called an agreement subtree. An exact algorithm for finding an agreement subtree is presented.

The agreement metric for labeled binary trees

Let S be a set of n objects. A binary tree of S is a binary tree whose leaves are labeled without repetition from S. The operation of pruning a tree T is that of removing some leaves from T and suppressing all inner vertices of degree 2 which are formed by this deletion. Given two trees T and U, an agreement tree is a tree that can be obtained from T as well as from U by pruning the fewest number of leaves from the two trees. A quadratic algorithm is presented for doing this and two metrics are defined based on agreement trees.

Using graph distance in object recognition

In this paper the concept of the distance between graphs is applied to the problem of recognizing objects. An unknown object is represented by a graph H and from the data base of possible solutions {G1, …, Gn} (also graphs) we select the graph Gi for which the distance to H is the smallest. The graph Gi is the recognition of H. The distance between graphs is defined in terms of edge rotation and deletion. It is shown that this distance defines a metric on the space of all graphs. The bounds for distance between two graphs are given in terms of their sizes and the size of their greatest common subgraph. It is proved that finding a distance between two graphs is an NP-complete problem even for planar graphs. An algorithm based on exhaustive search utilizing the linear algorithm by Hopcroft and Wong for recognizing isomorphism of planar graphs with some stopping criteria to maintain polynomial complexity, is possible.

Steiner intervals in graphs

Abstract LetG be a graph andu, v two vertices ofG. Then the interval fromu tov consists of all those vertices that lie on some shortestu — v path. LetS be a set of vertices in a connected graphG. Then the Steiner distancedG(S) ofS inG is the smallest number of edges in a connected subgraph ofG that containsS. Such a subgraph is necessarily a tree called a Steiner tree forS. The Steiner intervalIG(S) ofS consists of all those vertices that lie on some Steiner tree forS. LetS be ann-set of vertices ofG and suppose thatk ⩽ n. Then thek-intersection interval of S, denoted byIk(S) is the intersection of all Steiner intervals of allk-subsets ofS. It is shown that ifS = {u1, u2, ..., un} is a set ofn ⩾ 2 vertices of a graphG and if the 2-intersection interval ofS is nonempty andxeI2(S), then.d(S) = ∑ni = 1 =1 d(ui, x). It is observed that the only graphs for which the 2-intersection intervals of alln-sets,n ⩾ 4, are nonempty are stars. Moreover, for everyn ⩾ 4, those graphs with the property that the 3-intersection interval of everyn-set is nonempty are completely characterized. In general, ifn = 2k, those graphsG for whichIk(S) is nonempty for everyn-setS ofG are characterized.

On multipartite tournaments

Abstract An n -partite tournament, n ≥2, or multipartite tournament is an oriented graph obtained by orienting each edge of a complete n -partite graph. The cycle structure of multipartite tournaments is investigated and properties of vertices with maximum score are studied.

How to Choose the Best Twins

We consider a version of the secretary problem where each candidate has an identical twin. The aim, as in the classical problem, is to choose with the largest possible probability a top candidate, i.e., one of the best twins. We find an optimal stopping time for such a choice, the probability of success the optimal stopping time yields, and their asymptotic behavior.

On domination and reinforcement numbers in trees

The reinforcement number of a graph is the smallest number of edges that have to be added to a graph to reduce the domination number. We introduce the k-reinforcement number of a graph as the smallest number of edges that have to be added to a graph to reduce the domination number by k. We present an O(k^2n) dynamic programming algorithm for computing the maximum number of vertices that can be dominated using @c(G)-k dominators for trees. A corollary of this is a linear-time algorithm for computing the k-reinforcement number of a tree. We also discuss extensions and related problems.

A Ratio Inequality for Binary Trees and the Best Secretary

Let Tn be the complete binary tree of height n considered as the Hasse diagram of a poset with its root 1n as the maximum element. Define A(n; T) = m{S ⊆ Tn : 1n ∈ S, S ≅ T}m, and B(n; T) = m{S ⊆ Tn : 1n ∉ S, S ≅ T}m. In this note we prove that ***** insert equation here ***** for any fixed n and rooted binary trees T1, T2 such that T2 contains a subposet isomorphic to T1. We conjecture that the ratio A/B also increases with T for arbitrary trees. These inequalities imply natural behaviour of the optimal stopping time in a poset extension of the secretary problem.

The asymptotic plurality rule for molecular sequences

The asymptotic plurality rule, apl, is a consensus function which maps each profile P of length n (i.e., each sequence of n bases appearing at an aligned position of n molecules) to a set apl (P) of consensus results (i.e., ambiguity codes) that is a descriptive summary of P. Our main result is to characterize each consensus result X = apl(P) in terms of the frequencies with which the bases in P occur. We then use these characterizations to investigate features (e.g., strong consistency, length independence) of apl that researchers may find useful for the interpretation of apls consensus results.

The irregularity cost or sum of a graph

Abstract Working simultaneously in two teams [1,2], we have independently discovered essentially the same concept and many common results. As expected, each team used its own notation and terminology but the results are easily transformed between the two systems. We plan to publish our full papers separately, but present the results here.