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1.
Distance-based Invariants and Measures in Graphs
Aleksander Kelenc, 2019, doktorska disertacija

Opis: This doctoral dissertation is concerned with aspects on distance related topics in graphs. We study three main topics, namely a recently introduced measure called the Hausdorff distance of graphs and two new graph invariants - the edge metric dimension and the mixed metric dimension of graphs. All three topics are part of the metric graph theory since they are tightly connected with the basic concept of distance between two vertices of a graph. The Hausdorff distance is a relatively new measure of the similarity of graphs. The notion of the Hausdorff distance considers a special kind of common subgraph of the compared graphs and depends on the structural properties outside of the common subgraph. We study the Hausdorff distance between certain families of graphs that often appear in chemical graph theory. Next to a few results for general graphs, we determine formulae for the distance between paths and cycles. Previously, there was no known efficient algorithm for the problem of determining the Hausdorff distance between two trees, and in this dissertation we present a polynomial-time algorithm for it. The algorithm is recursive and it utilizes the divide and conquer technique. As a subtask it also uses a procedure that is based on the well-known graph algorithm for finding a maximum bipartite matching. The edge metric dimension is a graph invariant that deals with distinguishing the edges of a graph. Let $G=(V(G),E(G))$ be a connected graph, let $w \in V(G)$ be a vertex, and let $e=uv \in E(G)$ be an edge. The distance between the vertex $w$ and the edge $e$ is given by $d_G(e,w)=\min\{d_G(u,w),d_G(v,w)\}$. A vertex $w \in V(G)$ distinguishes two edges $e_1,e_2 \in E(G)$ if $d_G(w,e_1) \ne d_G(w,e_2)$. A set $S$ of vertices in a connected graph $G$ is an edge metric generator of $G$ if every two distinct edges of $G$ are distinguished by some vertex of $S$. The smallest cardinality of an edge metric generator of $G$ is called the edge metric dimension and is denoted by $dim_e(G)$. The concept of the edge metric dimension is new. We study its mathematical properties. We make a comparison between the edge metric dimension and the standard metric dimension of graphs while presenting some realization results concerning the two. We prove that computing the edge metric dimension of connected graphs is NP-hard and give some approximation results. Moreover, we present bounds and closed formulae for the edge metric dimension of several classes of graphs. The mixed metric dimension is a graph invariant similar to the edge metric dimension that deals with distinguishing the elements (vertices and edges) of a graph. A vertex $w \in V(G)$ distinguishes two elements of a graph $x,y \in E(G)\cup V(G)$ if $d_G(w,x) \ne d_G(w,y)$. A set $S$ of vertices in a connected graph $G$ is a mixed metric generator of $G$ if every two elements $x,y \in E(G) \cup V(G)$ of $G$, where $x \neq y$, are distinguished by some vertex of $S$. The smallest cardinality of a mixed metric generator of $G$ is called the mixed metric dimension and is denoted by $dim_m(G)$. In this dissertation, we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds. We also give results on the mixed metric dimension of certain families of graphs and present an upper bound with respect to the girth of a graph. Finally, we prove that the problem of determining the mixed metric dimension of a graph is NP-hard in the general case.
Ključne besede: Hausdorff distance, distance between graphs, graph algorithms, trees, graph similarity, edge metric dimension, edge metric generator, mixed metric dimension, metric dimension
Objavljeno: 03.08.2020; Ogledov: 415; Prenosov: 68
.pdf Celotno besedilo (800,48 KB)

2.
Wiener numbers of pericondensed benzenoid hydrocarbons
Sandi Klavžar, Ivan Gutman, Amal Rajapakse, 1997, izvirni znanstveni članek

Opis: Using a recently developed technique for the calculation of the Wiener number (W) of benzenoid systems, we determine explicit expressions for W for several homologous series of pericondensed benzenoid hydrocarbons. An elementary proof for the correctness of the used method is also included.
Ključne besede: mathematics, chemical graph theory, distance in graphs, Wiener number, benzenoids
Objavljeno: 05.07.2017; Ogledov: 715; Prenosov: 128
.pdf Celotno besedilo (16,93 MB)
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3.
Corroborating a modification of the Wiener index
Ivan Gutman, Janez Žerovnik, 2002, kratki znanstveni prispevek

Opis: In a recent work [Chem. Phys. Lett. 333 (2001) 319-321] Nikolić, Trinajstić, and Randie put forward a novel modification ▫$^m$▫W of the Wiener index. We now show that ▫$^m$▫W possesses the basic properties required by a topological index to be acceptable as a measure of the extent of branching of the carbon-atom skeleton of the respective molecule (and therefore to be a structure-descriptor, potentially applicable in QSPR and QSAR studies). In particular, if ▫$T_n$▫ is any n-vertex tree, different from the n-vertex path ▫$P_n$▫ and the n-vertex star ▫$S_n$▫, then mw(Pn) < mW(Tn) < mW(Sn). We also show how the concept of the modified Wiener index can be extended to weighted molecular graphs.
Ključne besede: graph theory, distance, molecular graphs, modified Wiener index, weigted modified Wiener index, branching, chemical graph theory
Objavljeno: 05.07.2017; Ogledov: 553; Prenosov: 57
.pdf Celotno besedilo (85,05 KB)
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4.
On acyclic colorings of direct products
Simon Špacapan, Aleksandra Tepeh, 2008, izvirni znanstveni članek

Opis: A coloring of a graph ▫$G$▫ is an acyclic coloring if the union of any two color classes induces a forest. It is proved that the acyclic chromatic number of direct product of two trees ▫$T_1$▫ and ▫$T_2$▫ equals ▫$\min\{ \Delta(T_1) + 1, \Delta(T_2) + 1\}$▫. We also prove that the acyclic chromatic number of direct product of two complete graphs ▫$K_m$▫ and ▫$K_n$▫ is ▫$mn-m-2$▫, where ▫$m \ge n \ge 4$▫. Several bounds for the acyclic chromatic number of direct products are given and in connection to this some questions are raised.
Ključne besede: mathematics, graph theory, coloring, acyclic coloring, distance-two coloring, direct product
Objavljeno: 31.03.2017; Ogledov: 386; Prenosov: 62
.pdf Celotno besedilo (142,13 KB)
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5.
A forbidden subgraph characterization of some graph classes using betweenness axioms
Manoj Changat, Anandavally K. Lakshmikuttyamma, Joseph Mathews, Iztok Peterin, Prasanth G. Narasimha-Shenoi, Geetha Seethakuttyamma, Simon Špacapan, 2013, izvirni znanstveni članek

Opis: Naj bo ▫$I_G(x,y)$▫ interval najkrajših ▫$x,y$▫-poti in ▫$J_G(x,y)$▫ interval induciranih ▫$x,y$▫-poti v povezanem grafu ▫$G$▫. Obravnavani so naslednji trije aksiomi vmesnosti za množico ▫$V$▫ in ▫$R: V times V rightarrow 2^V$▫: (i) ▫$x in R(u,y), y in R(x,v), x neq y, |R(u,v)|>2 Rightarrow x in R(u,v)$▫; (ii) ▫$x in R(u,v) Rightarrow R(u,x) cap R(x,v) = {x}$▫; (iii) ▫$x in R(u,y), y in R(x,v), x neq y, Rightarrow x in R(u,v)$▫. Karakteriziramo razred grafov, za katere ▫$I_G$▫ izpolnjuje (i), razred grafov, za katere ▫$J_G$▫ izpolnjuje (ii) in razred grafov, kjer oba ▫$I_G$▫ in ▫$J_G$▫ izpolnjujeta (iii). Karakterizacije so podane z prepovedanimi induciranimi podgrafi. Izkaže se, da je razred grafov, kjer ▫$I_G$▫ izpolnjuje (i), pravi podrazred razdaljno dednih grafov in da je razred, kjer ▫$J_G$▫ izpolnjuje (ii), pravi nadrazred razdaljno dednih grafov. Podani sta tudi aksiomatični karakterizaciji tetivnih in ptolomejskih grafov.
Ključne besede: matematika, teorija grafov, prepovedani podgrafi, inducirana pot, intervalna funkcija, aksiomi vmesnosti, tetivni grafi, razdaljno dedni grafi, mathematics, graph theory, forbidden subgraphs, induced path, interval function, betweenness axioms, chordal graphs, distance hereditary graphs
Objavljeno: 10.07.2015; Ogledov: 668; Prenosov: 84
URL Povezava na celotno besedilo

6.
Some Steiner concepts on lexicographic products of graphs
Bijo S. Anand, Manoj Changat, Iztok Peterin, Prasanth G. Narasimha-Shenoi, 2012, izvirni znanstveni članek

Opis: The smallest tree that contains all vertices of a subset ▫$W$▫ of ▫$V(G)$▫ is called a Steiner tree. The number of edges of such a tree is the Steiner distance of ▫$W$▫ and union of all Steiner trees of ▫$W$▫ form a Steiner interval. Both of them are described for the lexicographic product in the present work. We also give a complete answer for the following invariants with respect to the Steiner convexity: the Steiner number, the rank, the hull number, and the Carathéodory number, and a partial answer for the Radon number. At the end we locate and repair a small mistake from [J. Cáceres, C. Hernando, M. Mora, I. M. Pelayo, M. L. Puertas, On the geodetic and the hull numbers in strong product graphs, Comput. Math. Appl. 60 (2010) 3020--3031].
Ključne besede: teorija grafov, leksikografski produkt, Steinerjeva konveksnost, Steinerjeva množica, Steinerjeva razdalja, graph theory, lexicographic product, Steiner convexity, Steiner set, Steiner distance
Objavljeno: 10.07.2015; Ogledov: 644; Prenosov: 89
URL Povezava na celotno besedilo

7.
On a local 3-Steiner convexity
Boštjan Brešar, Tanja Gologranc, 2011, izvirni znanstveni članek

Opis: Za dani graf ▫$G$▫ je Steinerjev interval množice vozlišč ▫$W subset V(G)$▫ množica tistih vozlišč, ki ležijo na kakem Steinerjevem drevesu glede na ▫$W$▫. Množica ▫$U subset V(G)$▫ je ▫$g_3$▫-konveksna v ▫$G$▫, če Steinerjev interval poljubne trojice vozlišč iz ▫$U$▫ v celoti leži v ▫$U$▫. Henning, Nielsen in Oellermann (2009) so dokazali, da graf ▫$G$▫, v katerem so ▫$j$▫-krogle ▫$g_3$▫-konveksne za vsak ▫$j ge 1$▫, ne vsebuje hiše niti grafov dvojčkov ▫$C_4$▫ kot induciranih podgrafov in vsak cikel v ▫$G$▫ dolžine vsaj šest je dobro premostljiv. V tem članku dokažemo, da velja tudi obrat tega izreka, s čimer okarakteriziramo grafe z ▫$g_3$▫-konveksnimi kroglami.
Ključne besede: matematika, teorija grafov, Steinerjev interval, razdalja, dobra premostljivost, mathematics, graph theory, Steiner interval, distance, well-bridgeness
Objavljeno: 10.07.2015; Ogledov: 503; Prenosov: 26
URL Povezava na celotno besedilo

8.
Steiner intervals, geodesic intervals, and betweenness
Boštjan Brešar, Manoj Changat, Joseph Mathews, Iztok Peterin, Prasanth G. Narasimha-Shenoi, Aleksandra Tepeh, 2009, izvirni znanstveni članek

Opis: Koncept ▫$k$▫-Steinerjevih intervalov naravno posplošuje geodetske (binarne) intervale. Definiran je kot preslikava ▫$S: Vtimes cdots times V longrightarrow 2^V$▫, kjer je ▫$S(u_1, dots ,u_k)$▫ množica tistih vozlišč grafa ▫$G$▫, ki ležijo na kakem Steinerjevem drevesu glede na multimnožico ▫$W = {u_1, dots ,u_k}$▫ vozlišč iz ▫$G$▫. V tem članku za vsako naravno število ▫$k$▫ dokažemo karakterizacijo razreda tistih grafov, v katerih imajo vsi ▫$k$▫-Steinerjevi intervali t.i. lastnost unije, ki pravi, da ▫$S(u_1,ldots, u_k)$▫ sovpada z unijo geodetskih intervalov ▫$I(u_i,u_j)$▫ med vsemi pari vozlišč iz ▫$W$▫. Izkaže se, da tedaj, ko je ▫$k>3$▫, ta razred sovpada z razredom grafov, v katerih ▫$k$▫-Steinerjev interval zadošča aksiomu monotonosti(m), kot tudi z razredom grafov, v katerih ▫$k$▫-Steinerjev interval zadošča aksiomu (b2), ki sta pogoja iz teorije vmesnosti. In sicer preslikava ▫$S$▫ zadošča aksiomu (m), če iz ▫$x_1, dots ,x_k in S(u_1, dots ,u_k)$▫ sledi ▫$S(x_1, dots ,x_k) subseteq S(u_1, dots ,u_k)$▫; ter ▫$S$▫ zadošča (b2), če iz ▫$x in S(u_1,u_2, dots ,u_k)$▫ sledi ▫$S(x,u_2, dots ,u_k) subseteq S(u_1, dots ,u_k)$▫. V primeru ▫$k=3$▫ so ti trije razredi grafov različni in za razreda grafov, v katerih Steinerjev interval zadošča lastnosti unije oz. aksiomu monotonosti (m), dokažemo strukturni karakterizaciji. Prav tako predstavimo več delnih ugotovitev za razred grafov, v katerih 3-Steinerjev interval zadošča aksimu (b2), ki vodijo do domneve, da so to natanko tisti grafi, v katerih je vsak blok geodetski graf z diametrom 2.
Ključne besede: matematika, teorija grafov, Steinerjev interval, geodetski interval, razdalja, vmesnost, monotonost, bločni graf, mathematics, graph theory, Steiner interval, geodesic interval, distance, betweenness, monotonicity, block graph
Objavljeno: 10.07.2015; Ogledov: 539; Prenosov: 73
URL Povezava na celotno besedilo

9.
Many distances in planar graphs
Sergio Cabello, 2009

Opis: We show how to compute in ▫$O(n^{4/3} log^{1/3} n + n^{2/3}k^{2/3} log n)$▫ time the distance between $k$ given pairs of vertices of a planar graph $G$ with ▫$n$▫ vertices. This improves previous results whenever ▫$(n/log n)^{5/6} le k le n^2/log^6 n$▫. As an application, we speed up previous algorithms for computing the dilation of geometric planar graphs.
Ključne besede: matematika, teorija grafov, ravninski graf, razdalja, mathematics, graph theory, planar graph, distance
Objavljeno: 10.07.2015; Ogledov: 648; Prenosov: 59
URL Povezava na celotno besedilo

10.
Strongly distance-balanced graphs and graph products
Kannan Balakrishnan, Manoj Changat, Iztok Peterin, Simon Špacapan, Primož Šparl, Ajitha R. Subhamathi, 2008, izvirni znanstveni članek

Opis: A graph ▫$G$▫ is strongly distance-balanced if for every edge ▫$uv$▫ of ▫$G$▫ and every ▫$i ge 0$▫ the number of vertices ▫$x$▫ with ▫$d(x,u) = d(x,v)-1 = i$▫ equals the number of vertices ▫$y$▫ with ▫$d(y,v) = d(y,u)-1 = i$▫. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distance-balanced. It is also proved that connected components of the direct product of two bipartite graphs are strongly distance-balanced if and only if both factors are strongly distance-balanced. Additionally, a new characterization of distance-balanced graphs and an algorithm of time complexity ▫$O(mn)$▫ for their recognition, where m is the number of edges and ▫$n$▫ the number of vertices of the graph in question, are given.
Ključne besede: matematika, teorija grafov, razdaljno uravnoteženi grafi, mathematics, graph theory, distance-balanced grapha
Objavljeno: 10.07.2015; Ogledov: 934; Prenosov: 70
URL Povezava na celotno besedilo

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