arXiv subjects

Jelena Sedlar

Publications and source records attributed to Jelena Sedlar.

31 records · Page 2Linked to original sources

arXiv2022

Remarks on the vertex and the edge metric dimension of 2-connected graphs

The vertex (resp. edge) metric dimension of a graph G is the size of a smallest vertex set in G which distinguishes all pairs of vertices (resp. edges) in G and it is denoted by dim(G) (resp. edim(G)). The upper bounds dim(G) <= 2c(G) - 1 and edim(G) <= 2c(G)-1; where c(G) denotes the cyclomatic number of G, were established to hold for cacti without leaves distinct from cycles, and moreover all leafless cacti which attain the bounds were characterized. It was further conjectured that the same bounds hold for general connected graphs without leaves and this conjecture was supported by showing that the problem reduces to 2-connected graphs. In this paper we focus on Theta graphs, as the most simple 2-connected graphs distinct from cycle, and show that the the upper bound 2c(G) - 1 holds for both metric dimensions of Theta graphs and we characterize all Theta graphs for which the bound is attained. We conclude by conjecturing that there are no other extremal graphs for the bound 2c(G) - 1 in the class of leafless graphs besides already known extremal cacti and extremal Theta graphs mentioned here.

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arXiv2021

Remarks on the Local Irregularity Conjecture

A locally irregular graph is a graph in which the end-vertices of every edge have distinct degrees. A locally irregular edge coloring of a graph G is any edge coloring of G such that each of the colors induces a locally irregular subgraph of G. A graph G is colorable if it admits a locally irregular edge coloring. The locally irregular chromatic index of a colorable graph G, denoted by CHI'irr(G), is the smallest number of colors used by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all graphs, except odd length path, odd length cycle and a certain class of cacti, are colorable by 3 colors. As the conjecture is valid for graphs with large minimum degree and all non-colorable graphs are vertex disjoint cacti, we take direction to study rather sparse graphs. In this paper, we give a cactus graph B which contradicts this conjecture, i.e. CHI'irr(B) = 4. Nevertheless, we show that the conjecture holds for unicyclic graphs and cacti with vertex disjoint cycles.

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arXiv2021

Metric dimensions vs. cyclomatic number of graphs with minimum degree at least two

The vertex (resp. edge) metric dimension of a connected graph G; denoted by dim(G) (resp. edim(G)), is defined as the size of a smallest set S in V (G) which distinguishes all pairs of vertices (resp. edges) in G: Bounds dim(G) <= L(G)+2c(G) and edim(G) <= L(G) + 2c(G); where c(G) is the cyclomatic number in G and L(G) depends on the number of leaves in G, are known to hold for cacti and are conjectured to hold for general graphs. In leafless graphs it holds that L(G) = 0; so for such graphs the conjectured upper bound becomes 2c(G). In this paper, we show that the bound 2c(G) cannot be attained by leafless cacti, so the upper bound for such cacti decreases to 2c(G)-1, and we characterize all extremal leafless cacti for the decreased bound. We conjecture that the decreased bound holds for all leafless graphs, i.e. graphs with minimum degree at least two. We support this conjecture by showing that it holds for all graphs with minimum degree at least three and that it is suficient to show that it holds for all 2-connected graphs, and we also verify the conjecture for graphs of small order.

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arXiv2021

Vertex and edge metric dimensions of cacti

In a graph G; a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S: The cardinality of a smallest vertex (resp. edge) metric generator is the vertex (resp. edge) metric dimension of G: In [19] we determined the vertex (resp. edge) metric dimension of unicyclic graphs and that it takes its value from two consecutive integers. Therein, several cycle configurations were introduced and the vertex (resp. edge) metric dimension takes the greater of the two consecutive values only if any of these configurations is present in the graph. In this paper we extend the result to cactus graphs i.e. graphs in which all cycles are pairwise edge disjoint. We do so by defining a unicyclic subgraph of G for every cycle of G and applying the already introduced approach for unicyclic graphs which involves the configurations. The obtained results enable us to prove the cycle rank conjecture for cacti. They also yield a simple upper bound on metric dimensions of cactus graphs and we conclude the paper by conjecturing that the same upper bound holds in general.

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arXiv2021

Vertex and edge metric dimensions of unicyclic graphs

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) (resp. E(G)) is called the vertex (resp. edge) metric dimension of G. In [16] it was shown that both vertex and edge metric dimension of a unicyclic graph G always take values from just two explicitly given consecutive integers that are derived from the structure of the graph. A natural problem that arises is to determine under what conditions these dimensions take each of the two possible values. In this paper for each of these two metric dimensions we characterize three graph configurations and prove that it takes the greater of the two possible values if and only if the graph contains at least one of these configurations. One of these configurations is the same for both dimensions, while the other two are specific for each of them. This enables us to establish the exact value of the metric dimensions for a unicyclic graph and also to characterize when each of these two dimensions is greater than the other one.

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arXiv2020

Extremal mixed metric dimension with respect to the cyclomatic number

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V(G)U E(G) is called the mixed metric dimension of G, and it is denoted by mdim(G). In [12] it was conjectured that for a graph G with cyclomatic number c(G) it holds that mdim(G) <= L1(G) + 2c(G) where L1(G) is the number of leaves in G. It is already proven that the equality holds for all trees and more generally for graphs with edge-disjoint cycles in which every cycle has precisely one vertex of degree >= 3. In this paper we determine that for every Theta graph G, the mixed metric dimension mdim(G) equals 3 or 4, with 4 being attained if and only if G is a balanced Theta graph. Thus, for balanced Theta graphs the above inequality is also tight. We conclude the paper by further conjecturing that there are no other graphs, besides the ones mentioned here, for which the equality mdim(G) = L1(G) + 2c(G) holds.

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arXiv2020

Bounds on metric dimensions of graphs with edge disjoint cycles

In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it distinguishes E(G). In this paper these invariants are considered first for unicyclic graphs, and it is shown that the vertex and edge metric dimensions obtain values from two particular consecutive integers, which can be determined from the structure of the graph. In particular, as a consequence, we obtain that these two invariants can differ for at most one for a same unicyclic graph. Next we extend the results to graphs with edge disjoint cycles showing that the two invariants can differ at most by c, where c is the number of cycles in such a graph. We conclude the paper with a conjecture that generalizes the previously mentioned consequences to graphs with prescribed cyclomatic number c by claiming that the difference of the invariant is still bounded by c.

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arXiv2020

Mixed metric dimension of graphs with edge disjoint cycles

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G)[E(G) is called the mixed metric dimension of G. In this paper we first establish the exact value of the mixed metric dimension of a unicycic graph G which is derived from the structure of G. We further consider graphs G with edge disjoint cycles in which a unicyclic restriction Gi is introduced for each cycle Ci: Applying the result for unicyclic graph to each Gi then yields the exact value of the mixed metric dimension of such a graph G. The obtained formulas for the exact value of the mixed metric dimension yield a simple sharp upper bound on the mixed metric dimension, and we conclude the paper conjecturing that the analogous bound holds for general graphs with prescribed cyclomatic number.

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arXiv2017

On inverse Wiener interval problem of trees

The Wiener index W(G) of a simple connected graph G is defined as the sum of distances over all pairs of vertices in a graph. We denote by W[T_{n}] the set of all values of Wiener index for a graph from class T_{n} of trees on n vertices. The largest interval of contiguous integers (contiguous even integers in case of odd n) is denoted by W^{int}[T_{n}]. In this paper we prove that both sets are of the cardinality (1/6)n^3+O(n^2) in the case of even n, while in the case of odd n we prove that the cardinality of both sets equals (1/(12))n^3+O(n^2) solving thus two conjectures posed in literature.

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arXiv2013

Extremal unicyclic graphs with respect to additively weighted Harary index

In this paper we define cycle-star graph CSk,n-k to be a graph on n vertices consisting of the cycle of length k and n-k leafs appended to the same vertex of the cycle. Also, we define cycle-path graph CPk,n-k to be a graph on n vertices consisting of the cycle of length k and of path on n-k vertices whose one end is linked to a vertex on a cycle. We establish that cycle-star graph CS3,n-3 is the only maximal graph with respect to additively weighted Harary index among all unicyclic graphs on n vertices, while cycle-path graph CP3,n-3 is the only minimal unicyclic graph (here n must be at least 5). The values of additively weighted Harary index for extremal unicyclic graphs are established, so these values are the upper and the lower bound for the value of additively weighted Harary index on the class of unicyclic graphs on n vertices.

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arXiv2012

Remoteness, proximity and few other distance invariants in graphs

We establish maximal trees and graphs for the difference of average distance and proximity proving thus the corresponding conjecture posed in [4]. We also establish maximal trees for the difference of average eccentricity and remoteness and minimal trees for the difference of remoteness and radius proving thus that the corresponding conjectures posed in [4] hold for trees.

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arXiv2011

On augmented eccentric connectivity index of graphs and trees

In this paper we establish all extremal graphs with respect to augmented eccentric connectivity index among all (simple connected) graphs, among trees and among trees with perfect matching. For graphs that turn out to be extremal explicit formulas for the value of augmented eccentric connectivity index are derived.

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arXiv2011

Independent sets in chain cacti

In this paper chain cacti are considered. First, for two specific classes of chain cacti (orto-chains and meta-chains of cycles with h vertices) the recurrence relation for independence polynomial is derived. That recurrence relation is then used in deriving explicit expressions for independence number and number of maximum independent sets for such chains. Also, the recurrence relation for total number of independent sets for such graphs is derived. Finaly, the proof is provided that orto-chains and meta-chains are the only extremal chain cacti with respect to total number of independent sets (orto-chains minimal and meta-chains maximal).

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