arXiv subjects

Simon Brezovnik

Publications and source records attributed to Simon Brezovnik.

13 recordsLinked to original sources

arXiv2026

Upper bounds for double Roman domination and $[k]$-Roman domination of cylindrical graphs $C_m \Box P_n$

Roman-type domination parameters form an important class of graph invariants that model protection and resource allocation problems on networks. Among them, $[k]$-Roman domination provides a unified framework that generalizes Roman, double Roman, and higher-order variants. In this paper we investigate the $[k]$-Roman domination number of cylindrical grids $C_m\Box P_n$ and derive several new constructive upper bounds. Our approach combines three complementary techniques: linear periodic constructions, uniform ceiling-type labelings, and packing-based refinements. We first analyze the case $C_9\Box P_n$, where these three families of bounds can be compared explicitly and their relative efficiency is shown to depend on the parameter $k$. We then extend the linear constructions to cylindrical grids whose circumference is a multiple of one of the values $3,\dots,9$, obtaining a unified family of upper bounds for $C_{rt}\Box P_n$. Motivated by the asymptotic behavior of these estimates, we further derive general upper bounds depending only on the residue class of $m$ modulo $5$, which apply to all cylindrical grids. As a consequence, we obtain explicit estimates for the double Roman domination number $γ_{[2]R}(C_m\Box P_n)$ and compare the resulting multiple-based constructions with the residue-class bounds. This comparison shows that the residue-class construction becomes asymptotically superior for all sufficiently large admissible circumferences, while several exceptional small cases remain better covered by tailored constructions.

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arXiv2026

Further results on $[k]$-Roman domination on cylindrical grids $C_m \Box P_n$

In this paper, we study the $[k]$-Roman domination number of cylindrical graphs $C_m \Box P_n$. Our analysis begins with a general lower bound based on local neighborhood constraints, showing that $γ_{[k]R}(C_m\Box P_n) > (k+1)\left\lceil\frac{mn}{5}\right\rceil.$ By exploiting the connection between $[k]$-Roman domination and efficient domination, we characterize those cylindrical graphs whose optimal $[k]$-Roman domination number is realized by configurations with minimum possible local neighborhood weight. For fixed small values $m\in\{5,\ldots,8\}$, we construct explicit periodic $[k]$-Roman dominating functions that yield constructive upper bounds. These constructions are further refined using ceiling-type adjustments and reductions based on packing sets. A systematic comparison of the resulting bounds shows how their relative strength depends on the parameter $k$ and on the length of the path.

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arXiv2026

Resonance graphs that are daisy cubes: from hypercubes to independent sets via resonant sets

Let $G$ be a plane elementary bipartite graph whose infinite face is forcing. We provide a bijection between the set of maximal hypercubes of its resonance graph and the set of maximal resonant sets of $G$, which generalizes a main result in [MATCH Commun. Math. Comput. Chem. 68 (2012) 65-77], where $G$ was only considered as an elementary benzenoid graph without nice coronenes. For a special case when $G$ is a peripherally 2-colorable graph, it follows that there is a bijection between the set of maximal hypercubes of its resonance graph and the set of maximal independent sets of a tree that is the inner dual of $G$. We then show that the resonance graph of a plane bipartite graph $G$ is a daisy cube if and only if it is the simplex graph of the complement of a forest. Finally, we characterize trees with at most 5 maximal independent sets to determine daisy cubes that are simplex graphs of the complements of trees and having at most five maximal vertices.

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arXiv2026

$[k]$-Roman domination on cylindrical grids $C_m \Box P_n$

Roman domination and its higher-order extensions have attracted considerable attention due to their natural interpretation in terms of defensive resource allocation on networks. The recently introduced $[k]$-Roman domination framework unifies classical Roman, double, triple, and higher-strength protection schemes by allowing each fortified vertex to provide up to $k$ levels of support. In this paper, we investigate the $[k]$-Roman domination number $γ_{[k]R}(G)$ on cylindrical grids $C_m \Box P_n$. We relate $[k]$-Roman domination to efficient domination and show that for efficient graphs one has $γ_{[k]R}(G)=(k+1)γ(G)$; as a consequence, we obtain explicit values for broad families of toroidal grids and determine exactly when the cylindrical graphs $C_m\Box P_n$ admit an efficient dominating set. Building on these structural insights, we derive several upper bounds for $γ_{[k]R}(C_m \Box P_n)$ for small fixed values of $m$, accompanied by explicit labeling patterns that attain these bounds. All obtained bounds are systematically compared, revealing parameter ranges in which different constructions dominate depending on the value of $k$ and the length of the path. In addition, we present exact packing numbers for selected cylindrical graphs, which complement the domination results and enable further refinements via local weight reductions. Our results extend and unify known domination-type parameters on grid-like structures and highlight new regularities that emerge as the reinforcement strength increases.

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arXiv2024

Resonance graphs of plane bipartite graphs as daisy cubes

We characterize plane bipartite graphs whose resonance graphs are daisy cubes, and therefore generalize related results on resonance graphs of benzenoid graphs, catacondensed even ring systems, as well as 2-connected outerplane bipartite graphs. Firstly, we prove that if $G$ is a plane elementary bipartite graph other than $K_2$, then the resonance graph of $G$ is a daisy cube if and only if the Fries number of $G$ equals the number of finite faces of $G$. Next, we extend the above characterization from plane elementary bipartite graphs to plane bipartite graphs and show that the resonance graph of a plane bipartite graph $G$ is a daisy cube if and only if $G$ is weakly elementary bipartite such that each of its elementary component $G_i$ other than $K_2$ holds the property that the Fries number of $G_i$ equals the number of finite faces of $G_i$. Along the way, we provide a structural characterization for a plane elementary bipartite graph whose resonance graph is a daisy cube, and show that a Cartesian product graph is a daisy cube if and only if all of its nontrivial factors are daisy cubes.

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arXiv2024

On the Wiener-like root-indices of graphs

In this paper, we examine roots of graph polynomials where those roots can be considered as structural graph measures. More precisely, we prove analytical results for the roots of certain modified graph polynomials and also discuss numerical results. As polynomials, we use, e.g., the Hosoya, the Schultz, and the Gutman polynomial which belong to an interesting family of degree-distance-based graph polynomials; they constitute so-called counting polynomials with non-negative integers as coefficients and the roots of their modified versions have been used to characterize the topology of graphs. Our results can be applied for the quantitative characterization of graphs. Besides analytical results, we also investigate other properties of those measures such as their degeneracy which is an undesired aspect of graph measures. It turns out that the measures representing roots of graph polynomials possess high discrimination power on exhaustively generated trees, which outperforms standard versions of these indices. Furthermore, a new measure is introduced that allows us to compare different topological indices in terms of structure sensitivity and abruptness.

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arXiv2024

The 2-rainbow domination number of Cartesian product of cycles

A $k$-rainbow dominating function ($k$RDF) of $G$ is a function that assigns subsets of $ \{1,2,...,k\}$ to the vertices of $G$ such that for vertices $v$ with $f(v)=\emptyset $ we have $\bigcup\nolimits_{u\in N(v)}f(u)=\{1,2,...,k\}$. The weight $w(f)$ of a $k$RDF $f$ is defined as $w(f)=\sum_{v\in V(G)}\left\vert f(v)\right\vert $. The minimum weight of a $k$RDF of $G$ is called the $k$-rainbow domination number of $G$, which is denoted by $γ_{rk}(G)$. In this paper, we study the 2-rainbow domination number of the Cartesian product of two cycles. Exact values are given for a number of infinite families and we prove lower and upper bounds for all other cases.

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arXiv2023

Outerplane bipartite graphs with isomorphic resonance graphs

We present novel results related to isomorphic resonance graphs of 2-connected outerplane bipartite graphs. As the main result, we provide a structure characterization for 2-connected outerplane bipartite graphs with isomorphic resonance graphs. Moreover, two additional characterizations are expressed in terms of resonance digraphs and via local structures of inner duals of 2-connected outerplane bipartite graphs, respectively.

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arXiv2020

General cut method for computing Szeged-like topological indices with applications to molecular graphs

Szeged, PI and Mostar indices are some of the most investigated distance-based molecular descriptors. Recently, many different variations of these topological indices appeared in the literature and sometimes they are all together called Szeged-like topological indices. In this paper, we formally introduce the concept of a general Szeged-like topological index, which includes all mentioned indices and also infinitely many other topological indices that can be defined in a similar way. As the main result of the paper, we provide a cut method for computing a general Szeged-like topological index for any strength-weighted graph. This greatly generalizes various methods known for some of the mentioned indices and therefore rounds off such investigations. Moreover, we provide applications of our main result to benzenoid systems, phenylenes, and coronoid systems, which are well-known families of molecular graphs. In particular, closed-form formulas for some subfamilies of these graphs are deduced.

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arXiv2020

Grundy domination and zero forcing in regular graphs

Given a finite graph $G$, the maximum length of a sequence $(v_1,\ldots,v_k)$ of vertices in $G$ such that each $v_i$ dominates a vertex that is not dominated by any vertex in $\{v_1,\ldots,v_{i-1}\}$ is called the Grundy domination number, $γ_{\rm gr}(G)$, of $G$. A small modification of the definition yields the Z-Grundy domination number, which is the dual invariant of the well-known zero forcing number. In this paper, we prove that $γ_{\rm gr}(G) \geq \frac{n + \lceil \frac{k}{2} \rceil - 2}{k-1}$ holds for every connected $k$-regular graph of order $n$ different from $K_{k+1}$ and $\bar{2C_4}$. The bound in the case $k=3$ reduces to $γ_{\rm gr}(G) \geq \frac{n}{2}$, and we characterize the connected cubic graphs with $γ_{\rm gr}(G)=\frac{n}{2}$. If $G$ is different from $K_4$ and $K_{3,3}$, then $\frac{n}{2}$ is also an upper bound for the zero forcing number of a connected cubic graph, and we characterize the connected cubic graphs attaining this bound.

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arXiv2019

Resonance graphs of catacondensed even ring systems

A catacondensed even ring system (shortly CERS) is a simple bipartite 2-connected outerplanar graph with all vertices of degree 2 or 3. In this paper, we investigate the resonance graphs (also called $Z$-transformation graphs) of CERS and firstly show that two even ring chains are resonantly equivalent iff their resonance graphs are isomorphic. As the main result, we characterize CERS whose resonance graphs are daisy cubes. In this way, we greatly generalize the result known for kinky benzenoid graphs. Finally, some open problems are also presented.

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arXiv2019

Resonantly Equivalent Catacondensed Even Ring Systems

In this paper we generalize the binary coding procedure of perfect matchings from catacondensed benzenoid graphs to catacondensed even ring systems (also called cers). Next, we study cers with isomorphic resonance graphs. For this purpose, we define resonantly equivalent cers. Finally, we investigate cers whose resonance graphs are isomorphic to the resonance graphs of catacondensed benzenoid graphs. As a consequence we show that for each phenylene there exists a catacondensed benzenoid graph such that their resonance graphs are isomorphic.

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arXiv2018

New methods for calculating the degree distance and the Gutman index

In the paper we develop new methods for calculating the two well-known topological indices, the degree-distance and the Gutman index. Firstly, we prove that the Wiener index of a double vertex-weighted graph can be computed from the Wiener indices of weighted quotient graphs with respect to a partition of the edge set that is coarser than $Θ^*$-partition. This result immediately gives a method for computing the degree-distance of any graph. Next, we express the degree-distance and the Gutman index of an arbitrary phenylene by using its hexagonal squeeze and inner dual. In addition, it is shown how these two indices of a phenylene can be obtained from the four quotient trees. Furthermore, reduction theorems for the Wiener index of a double vertex-weighted graph are presented. Finally, a formula for computing the Gutman index of a partial Hamming graph is obtained.

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