Graafiülesannete lahendamise keskkond
Date
2013
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Tartu Ülikool
Abstract
Aines „Diskreetse matemaatika elemendid“ (MTAT.05.109) kasutatakse õpetamisel mitmesuguseid õpiprogramme. Hetkel ei ole kasutusel programmi graafiteooria õpetamiseks. Käesoleva bakalaureusetöö eesmärgiks oligi luua graafiülesannete lahendamise keskkonna prototüüp.
Keskkond koosneb kahest programmist: „graafiülesannete koostaja“ ja „graafiülesannete lahendaja“. Koostamisprogrammi abil saab koostada ülesandekogusid. Lahendusprogrammis saab kogude põhjal luua lahendusfaile ja nendes ülesandeid lahendada. Lahendusfaile saab koostamisprogrammi abil ülevaatlikult kontrollida.
Hetkel on keskkonnas võimalik luua ja lahendada kuut tüüpi ülesandeid:
1. naabrusmaatriksile vastava graafi joonistamine,
2. graafile vastava naabrusmaatriksi konstrueerimine,
3. täisgraafi joonistamine,
4. etteantud graafi täiendgraafi joonistamine,
5. tipuastmejärjendile vastava graafi joonistamine,
6. graafile vastava tipuastmejärjendi konstrueerimine.
Nii ülesannete loomisel kui lahendamisel läheb vaja kolme paneeli: graafi joonistamise paneel, maatriksi konstrueerimise paneel ja tipuastmejärjendi konstrueerimise paneel.
Ülesande loomisel on kitsendused selle jaoks, et ei saaks luua vigaseid ülesandeid. Lahendamisel kontrollitakse erinevat tüüpi vigade esinemist.
Üheks huvitavamaks probleemiks oli see, kuidas programmis kontrollida, kas järjendi põhjal saab graafi joonistada või mitte. Lahenduseks oli Erdős-Gallai teoreemi kasutamine.
Tulevikus on võimalik keskkonda lisada suunatud graafide tugi ning lisaks sellele ka erinevaid ülesandetüüpe.
In Elements of Discrete Mathematics (MTAT.05.109) course various programs are used for teaching. Currently there is no program for creating and solving graph theory exercises. The purpose of this Bachelor’s thesis was to create a prototype for graph theory exercise environment. Only undirected graphs are supported in the prototype and there are six types of exercises: 1. drawing a graph based on a given adjacency matrix, 2. constructing an adjacency matrix based on a given graph, 3. drawing a complete graph, 4. drawing a complement graph of a given graph, 5. constructing a degree sequence based on a given graph, 6. drawing a graph based on a given degree sequence. Three panels are used in different combinations for these six exercises: graph panel, matrix panel and degree sequence panel. They are used to create and solve exercises. When creating an exercise there are constraints to prevent incorrect or empty exercises. When solving the exercises the program checks for different types of errors. All errors are recorded in the file and can be reviewed later. One of the problems encountered was that a suitable algorithm had to be found to check whether a degree sequence is graphical or not. This problem was solved using the Erdős-Gallai theorem. In the future it is possible to add support for directed graphs. New exercise types can also be added.
In Elements of Discrete Mathematics (MTAT.05.109) course various programs are used for teaching. Currently there is no program for creating and solving graph theory exercises. The purpose of this Bachelor’s thesis was to create a prototype for graph theory exercise environment. Only undirected graphs are supported in the prototype and there are six types of exercises: 1. drawing a graph based on a given adjacency matrix, 2. constructing an adjacency matrix based on a given graph, 3. drawing a complete graph, 4. drawing a complement graph of a given graph, 5. constructing a degree sequence based on a given graph, 6. drawing a graph based on a given degree sequence. Three panels are used in different combinations for these six exercises: graph panel, matrix panel and degree sequence panel. They are used to create and solve exercises. When creating an exercise there are constraints to prevent incorrect or empty exercises. When solving the exercises the program checks for different types of errors. All errors are recorded in the file and can be reviewed later. One of the problems encountered was that a suitable algorithm had to be found to check whether a degree sequence is graphical or not. This problem was solved using the Erdős-Gallai theorem. In the future it is possible to add support for directed graphs. New exercise types can also be added.