20.500.12556/DKUM-54644 Robne in geodetske množice v grafih Boundary and geodetic sets in graphs V magistrskem delu so obravnavane lastnosti in povezave med posameznimi robnimi množicami grafa, ki jih sestavljajo robna, ekscentrična, periferna, konturna in ekstremna vozlišča grafa. Zanimale nas bodo predvsem povezave med robnimi in geodetskimi množicami grafa, posebej se bomo posvetili preučevanju konturne množice grafa. V prvem poglavju so zapisani osnovni pojmi in definicije iz teorije grafov, ki jih bomo potrebovali v nadaljevanju. V drugem poglavju definiramo tipe robnih množic, navedemo osnovne lastnosti le-teh in dokažemo dva realizacijska izreka, ki govorita o obstoju poljubnega grafa pri podanih kardinalnostih različnih skupin robnih množic. V tretjem poglavju navedemo rezultate, ki pravijo, da je konturna množica tetivnih, razdaljno hereditarnih, 3-SDH in HHD-prostih grafov geodetska množica. Obravnavamo tudi konturno množico dvodelnih grafov in dokažemo, da za vsak diameter $kgeq 8$ obstaja dvodelni graf, katerega konturna množica ni geodetska. V zadnjem razdelku obravnavamo konturne in geodetske množice delnih kock. In this master thesis we are dealing with the properties and relationships between boundary sets of a graph consisting of boundary, eccentric, peripheral, contour and extreme vertices of a graph. We study mostly the relationship between boundary sets and geodetic sets of an arbitrary graph, in particular we focus on the contour set of a graph. The first chapter introduces some basic definitions from graph theory that are needed for further understanding of the subject. In the second chapter we define different types of boundary sets and obtain some basic structural properties. We prove two realization theorems, which are referring to the existence of a graph, where cardinalities of different groups of boundary sets are given. In the third section we present results stating that the contour set of chordal, distance hereditary, 3-SDH and HHD-free graphs is the geodetic set. We also focus on the contour set of bipartite graphs and prove that for every diameter $kgeq 8$ there exists a bipartite graph, whose contour set is not a geodetic set. In the last section we study contour and geodetic sets of partial cubes. robne množice geodetska množica konturna množica boundary sets geodetic set contour set true true false [V. Lebar] Slovenski jezik Angleški jezik Magistrsko delo/naloga 2015-09-08 11:41:16 2015-11-05 15:22:54 2022-06-06 03:06:16 2021-05-12 03:18:21 2015 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17(043.2) 21693192 URN:SI:UM:DK:ODME7TH4 MAG_Lebar_Vesna_2015.pdf MAG_Lebar_Vesna_2015.pdf 1 6AA539DE23FDC1813E552331C4345C5B 7bdc357bfe9b5f211f3560c4472b53eab9651217769fc76006e86b90403ebd72 a8e0bca7-7c07-11eb-bb7a-00155d0001ca https://dk.um.si/Dokument.php?lang=slv&id=80575 Fakulteta za naravoslovje in matematiko 0 0 0