As a generalization of the concept metric basis, this article introduces notion $k$-metric basis in graphs. Given connected graph $G=(V,E)$, set $S\subseteq V$ is said to be generator for $G$ if elements any pair different vertices are distinguished by at least $k$ $S$, i.e., two $u,v\in V$, there exist $w_1,w_2,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$ every $i\in \{1,...,k\}$. A minimum cardinality called and its dimension $G$. dimensional largest integer exists We give necessary...

A Roman dominating function on a graph G is f:V(G) → {0,1,2} satisfying the condition that every vertex u for which f(u) = 0 adjacent to at least one v f(v) 2. The weight of f sum, ΣuV(G) f(u), weights vertices. domination number minimum in G. total with additional property subgraph induced by set all vertices positive has no isolated vertex. We establish lower and upper bounds number. relate parameters, including number,

The mutual-visibility problem in a graph G asks for the cardinality of largest set vertices S⊆V(G) so that any two x,y∈S there is shortest x,y-path P all internal are not S. This also said as x,y visible with respect to S, or S-visible short. Variations this known, based on extension visibility property and/or outside Such variations called total, outer and dual problems. work focused studying corresponding four parameters graphs diameter two, throughout showing bounds closed formulae these...

Let G be a graph and X⊆V(G). Then X is mutual-visibility set if each pair of vertices from connected by geodesic with no internal vertex in X. The number μ(G) the cardinality largest set. In this paper, strong product graphs investigated. As tool for this, total sets are introduced. Along way, basic properties such presented. (total) products bounded below two ways, determined exactly grids arbitrary dimension. Strong prisms studied separately couple tight bounds their given.

If X is a subset of vertices graph G, then u and v are X-visible if there exists shortest u,v-path P such that V(P)∩X⊆{u,v}. each two from X-visible, mutual-visibility set. The number G the cardinality largest set has been already investigated. In this paper variety problems introduced based on which natural pairs required to be X-visible. This yields total, dual, outer numbers. We first show these invariants related other classical number, we prove three newly computationally difficult....

A subset S of vertices a graph G is general position set if no shortest path in contains three or more S. In this paper, we generalise problem M. Gardner to theory by introducing the lower number gp -(G) G, which smallest maximal G.We show that = 2 and only universal line determine for several classes Di Stefano, Klavžar, Krishnakumar, Tuite Yero graphs, including Kneser graphs K(n, 2), complete Cartesian direct products two graphs.We also prove realisation results involving number, geodetic...

Abstract The general position number $$\mathrm{gp}(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>gp</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> of a connected graph G is the cardinality largest set S vertices such that no three distinct from lie on common geodesic; sets are refereed to as gp-sets . cylinders $$P_r\,\square \,C_s$$...

Given an ordered partition Π={P1,P2,…,Pt} of the vertex set V a connected graph G=(V,E), representation v∈V with respect to Π is vector r(v|Π)=(d(v,P1),d(v,P2),…,d(v,Pt)), where d(v,Pi) represents distance between v and Pi. A resolving G if different vertices have representations, i.e., for every pair u,v∈V, r(u|Π)≠r(v|Π). The dimension minimum number sets in any G. In this paper we obtain several tight bounds on trees.

Abstract The general position number gp( G ) of a connected graph is the cardinality largest set S vertices such that no three pairwise distinct from lie on common geodesic. It proved ≥ ω ( SR ), where strong resolving , and its clique number. That bound sharp demonstrated with numerous constructions including for instance direct products complete graphs different families products, generalized lexicographic rooted product graphs. For it ⊠ H )gp( asked whether equality holds arbitrary ....

Abstract Let G be a graph. Assume that to each vertex of set vertices $S\subseteq V(G)$ robot is assigned. At stage one can move neighbouring vertex. Then S mobile general position if there exists sequence moves the robots such all are visited while maintaining property at times. The number cardinality largest . We give bounds on and determine exact values for certain common classes graphs, including block rooted products, unicyclic Kneser graphs $K(n,2)$ line complete graphs.

A set $D$ of vertices a graph $G$ is total dominating if every vertex adjacent to at least one in $D$. The called co-independent $V(G)\setminus D$ an independent and has vertex. minimum cardinality any denoted by $\gamma_{t,coi}(G)$. In this paper, we show that, for tree $T$ order $n$ diameter three, $n-\beta(T)\leq \gamma_{t,coi}(T)\leq n-|L(T)|$ where $\beta(T)$ the maximum $L(T)$ leaves $T$. We also characterize families trees attaining extremal bounds above that differences between value...