Well-structured Classes Research Articles

Large independent sets in triangle-free cubic graphs: beyond planarity

The _independence ratio_ of a graph is the ratio of the size of its largest independent set to its number of vertices. Trivially, the independence ratio of a k-colorable graph is at least $1/k$ as each color class of a k-coloring is an independent set. However, better bounds can often be obtained for well-structured classes of graphs. In particular, Albertson, Bollobás and Tucker conjectured in 1976 that the independence ratio of every triangle-free subcubic planar graph is at least $3/8$. The conjecture was proven by Heckman and Thomas in 2006, and the ratio is best possible as there exists a cubic triangle-free planar graph with 24 vertices and the independence number equal to 9. The present article removes the planarity assumption. However, one needs to introduce an additional assumption since there are known to exist six 2-connected (non-planar) triangle-free subcubic graphs with the independence ratio less than $3/8$. Bajnok and Brinkmann conjectured that every 2-connected triangle-free subcubic graph has the independence ratio at least $3/8$ unless it is one of the six exceptional graphs. Fraughnaugh and Locke proposed a stronger conjecture: every triangle-free subcubic graph that does not contain one of the six exceptional graphs as a subgraph has independence ratio at least $3/8$. The authors prove these two conjectures, which implies in particular the result by Heckman and Thomas.

Investigating Student Sustained Attention in a Guided Inquiry Lecture Course Using an Eye Tracker

This study investigated the belief that student attention declines after the first 10 to 15 min of class by analyzing vigilance decrement in a guided inquiry physical science course. We used Tobii Glasses, a portable eye tracker, to record student gaze during class sessions. Undergraduate students (n = 17) representative of course demographics (14 female, 3 male) wore the eye tracker during 70-min classes (n = 84) or 50-min classes (n = 26). From the gaze point and fixation data, we coded participant attention as either on-task or off-task for every second of data. This analysis resulted in a percentage of vigilance time on task for each minute as well as the amount of time that participants spent looking in various locations during the class sessions. Participants exhibited on-task vigilance percentages starting with 67% at the start of class and rising to an average of above 90% on-task vigilance at the 7 to 9-min mark with minor fluctuation. Contrary to the belief that attention declines rapidly during a class, the participants on-task spans were larger and more numerous than their off-task spans. These results seem to support the conclusion that well-structured classes punctuated by student-student and instructor-student interactions can be an effective method of maintaining student attention vigilance for entire class sessions, not just the first 10 min.

When an optimal dominating set with given constraints exists

A dominating set is a set S of vertices in a graph such that every vertex not in S is adjacent to a vertex in S. In this paper, we consider the set of all optimal (i.e. smallest) dominating sets S, and ask of the existence of at least one such set S with given constraints. The constraints say that the number of neighbors in S of a vertex inside S must be in a given set ρ, and the number of neighbors of a vertex outside S must be in a given set σ. For example, if ρ is [1,k], and σ is the nonnegative integers, this corresponds to “[1,k]-domination.”First, we consider the complexity of recognizing whether an optimal dominating set with given constraints exists or not. We show via two different reductions that this problem is NP-hard for certain given constraints. This, in particular, answers a question of [M. Chellali et al., [1,2]-dominating sets in graphs, Discrete Applied Mathematics 161 (2013) 2885–2893] regarding the constraint that the number of neighbors in the set be upper-bounded by 2. We also consider the corresponding question regarding “total” dominating sets.Next, we consider some well-structured classes of graphs, including permutation and interval graphs (and their subfamilies), and determine exactly the smallest k such that for all graphs in that family an optimal dominating set exists where every vertex is dominated at most k times. We also consider the problem for trees (with implications for chordal and comparability graphs) and graphs with bounded “asteroidal number”.

Degree constrained 2-partitions of semicomplete digraphs

A 2-partition of a digraph D is a partition (V1,V2) of V(D) into two disjoint non-empty sets V1 and V2 such that V1∪V2=V(D). A semicomplete digraph is a digraph with no pair of non-adjacent vertices. We consider the complexity of deciding whether a given semicomplete digraph has a 2-partition such that each part of the partition induces a (semicomplete) digraph with some specified property. In [4] and [5] Bang-Jensen, Cohen and Havet determined the complexity of 120 such 2-partition problems for general digraphs. Several of these problems are NP-complete for general digraphs and thus it is natural to ask whether this is still the case for well-structured classes of digraphs, such as semicomplete digraphs. This is the main topic of the paper. More specifically, we consider 2-partition problems where the set of properties are minimum out-, minimum in- or minimum semi-degree. Among other results we prove the following:•For all integers k1,k2≥1 and k1+k2≥3 it is NP-complete to decide whether a given digraph D has a 2-partition (V1,V2) such that D〈Vi〉 has out-degree at least ki for i=1,2.•For every fixed choice of integers α,k1,k2≥1 there exists a polynomial algorithm for deciding whether a given digraph of independence number at most α has a 2-partition (V1,V2) such that D〈Vi〉 has out-degree at least ki for i=1,2.•For every fixed integer k≥1 there exists a polynomial algorithm for deciding whether a given semicomplete digraph has a 2-partition (V1,V2) such that D〈V1〉 has out-degree at least one and D〈V2〉 has in-degree at least k.•It is NP-complete to decide whether a given semicomplete digraph D has a 2-partition (V1,V2) such that D〈Vi〉 is a strong tournament.

Stanley sequences with odd character

Given a set of integers containing no 3-term arithmetic progressions, one constructs a Stanley sequence by choosing integers greedily without forming such a progression. Independent Stanley sequences are a well-structured class of Stanley sequences with two main parameters: the character $\lambda(A)$ and the repeat factor $\rho(A)$. Rolnick conjectured that for every $\lambda \in \mathbb{N}_0\backslash\{1, 3, 5, 9, 11, 15\}$, there exists an independent Stanley sequence $S(A)$ such that $\lambda(A) =\lambda$. This paper demonstrates that $\lambda(A) \not\in \{1, 3, 5, 9, 11, 15\}$ for any independent Stanley sequence $S(A)$.

Transforming RDB schema into well-structured OODB schema

When transforming relational database (RDB) schema into object-oriented database(OODB) schema, much effort was put on examining key and inclusion dependency (ID) constraints to identify class and establish inheritance and association between classes. However, in order to further remove the original data redundancy and update anomaly, multi-valued dependency (MVD) should also be examined. In this paper, we discuss class structures and define well-structured classes. Based on MVDs, a theorem is given transforming a relation schema into a well-structured class. To transform RDB schema into OODB schema, a composition process simplifying the input RDB schema and an algorithm transforming the simplified RDB schema into well-structured OODB classes are developed.

Hybrid forms of Reed-Muller expansions

The large well-structured class of Reed–Muller forms–which combine both normal and dual forms of basic operations and can be described by either Kronecker or pseudo-Kronecker transforms–is considered. It is demonstrated that these hybrid forms can be assembled into sets of expansions that are each equivalent to the set of normal forms. Corresponding members in each set involve the same form of transform matrix but require modification by means of a particular additive constant vector. Methods are described for deriving these constants so that existing design methods may be adapted to operate with these hybrid forms. A modular form of tree circuit is employed as a convenient means of describing and analysing the structure of these hybrid forms, and this is also considered as an alternative form of circuit synthesis.