The notion of resolving sets in a graph was introduced by Slater (1975) and Harary and Melter (1976) as a way of uniquely identifying every vertex in a graph A set of vertices in a graph is a resolving set if for any pair of vertices x and y there is a vertex in the set which has distinct distances to x and y A smallest resolving set in a graph is called a metric basis and its size, the metric dimension of the graph The problem of computing the metric dimension of a graph is a well-known NP-hard problem and while it was known to be polynomial time solvable on trees, it is only recently that efforts have been made to understand its computational complexity on various restricted graph classes In recent work, Foucaud et al (2015) showed that this problem is NP-complete even on interval graphs They complemented this result by also showing that it is fixed-parameter tractable (FPT) parameterized by the metric dimension of the graph In this work, we show that this FPT result can in fact be extended to all graphs of bounded tree-length This includes well-known classes like chordal graphs, AT-free graphs and permutation graphs We also show that this problem is FPT parameterized by the modular-width of the input graph

Metric dimension of bounded width graphs

We survey known results on enhanced power graphs of ﬁnite groups. Open problems, questions and suggestions for future work are also included.

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A survey on enhanced power graphs of finite groups

The problem of finding the largest connected subgraph of a given undirected host graph, subject to constraints on the maximum degree $\Delta$ and the diameter $D$, was introduced in \cite{maxddbs}, as a generalization of the Degree-Diameter Problem. A case of special interest is when the host graph is a common parallel architecture. Here we discuss the case when the host graph is a $k$-dimensional mesh. We provide some general bounds for the order of the largest subgraph in arbitrary dimension $k$, and for the particular cases of $k=3, \Delta = 4$ and $k=2, \Delta = 3$, we give constructions that result in sharper lower bounds.

The Maximum Degree-and-Diameter-Bounded Subgraph in the Mesh

Let G be a connected graph and u, v and w vertices of G. Then w is said to resolve u and v if the distance from u to w does not equal the distance from v to w. If there is either a shortest u-w path that contains v or a shortest v-w path that contains u, then we say that w strongly resolves u and v. A set W of vertices of G is a resolving set (strong resolving set), if every pair of vertices of G is resolved (respectively, strongly resolved) by some vertex of W. A smallest resolving set (strong resolving set) of a graph is called a basis (respectively, a strong basis) and its cardinality, denoted β(G) (respectively, βs(G)), the metric dimension (respectively, the strong dimension) of G. The threshold dimension (respectively, threshold strong dimension) of a graph G, denoted τ(G) (respectively, τs(G)), is the smallest metric dimension (respectively, strong dimension) among all graphs having G as a spanning subgraph. We survey results on the threshold dimension and threshold strong dimension and contrast these invariants. The paper concludes with several open problems for these parameters.

The Threshold Dimension and Threshold Strong Dimension of a Graph: A Survey

The outer multiset dimension $$\textrm{dim}_\textrm{ms}(G)$$ of a graph G is the cardinality of a smallest set of vertices that uniquely recognize all the vertices outside this set by using multisets of distances to the set. It is proved that $$\textrm{dim}_\textrm{ms}(G) = n(G) - 1$$ if and only if G is a regular graph with diameter at most 2. Graphs G with $$\textrm{dim}_\textrm{ms}(G)=2$$ are described and recognized in polynomial time. A lower bound on the lexicographic product of G and H is proved when H is complete or edgeless, and the extremal graphs are determined. It is proved that $$\textrm{dim}_\textrm{ms}(P_s\,\square \, P_t) = 3$$ for $$s\ge t\ge 2$$ . 

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Further Contributions on the Outer Multiset Dimension of Graphs

The multiset dimension was introduced by Rinovia Simanjuntak et al. as a variation of metric dimension. In this problem, the representation of a vertex v with respect to a resolving set W is expressed as a multiset of distances between v and all vertices in W , including their multiplicities. The multiset dimension is defined to be the minimum cardinality of the resolving set. Clearly, this is at least the metric dimension of a graph. In this paper, we study the properties of the multiset dimension of graphs.

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Some properties of the multiset dimension of graphs

Distance-irregular labeling was introduced by Slamin, and in his paper he determined the distance-irregular labeling for cycles and wheels of length {0, 1, 2, 5} mod 8. In this paper, we complete his results for cycles and wheels in general and prove the conjecture regarding the distance irregularity strength on wheels. We also show the general relation of the distance irregularity strength between a distance-irregular graph G and the graph G + K1. Finally, we determine the distance irregularity strength of m-book graphs Bm.

On distance-irregular labelings of cycles and wheels

The threshold strong dimension of a graph

Let $G$ be a connected graph and $u,v$ and $w$ vertices of $G$. Then $w$ is said to {\em strongly resolve} $u$ and $v$, if there is either a shortest $u$-$w$ path that contains $v$ or a shortest $v$-$w$ path that contains $u$. A set $W$ of vertices of $G$ is a {\em strong resolving set} if every pair of vertices of $G$ is strongly resolved by some vertex of $W$. A smallest strong resolving set of a graph is called a {\em strong basis} and its cardinality, denoted $\beta_s(G)$, the {\em strong dimension} of $G$. The {\em threshold strong dimension} of a graph $G$, denoted $\tau_s(G)$, is the smallest strong dimension among all graphs having $G$ as spanning subgraph. A graph whose strong dimension equals its threshold strong dimension is called $\beta_s$-{\em irreducible}. In this paper we establish a geometric characterization for the threshold strong dimension of a graph $G$ that is expressed in terms of the smallest number of paths (each of sufficiently large order) whose strong product admits a certain type of embedding of $G$. We demonstrate that the threshold strong dimension of a graph is not equal to the previously studied threshold dimension of a graph. Graphs with strong dimension $1$ and $2$ are necessarily $\beta_s$-irreducible. It is well-known that the only graphs with strong dimension $1$ are the paths. We completely describe graphs with strong dimension $2$ in terms of the strong resolving graphs introduced by Oellermann and Peters-Fransen. We obtain sharp upper bounds for the threshold strong dimension of general graphs and determine exact values for this invariant for certain subclasses of trees.

The Threshold Strong Dimension of a Graph.

A maximum degree-diameter bounded subgraph problem can be seen as a degree-diameter problem restricted to certain host graphs. In this paper, we investigate the MaxDDBS problem when the host graph is a butterfly network. We give constructive lower bounds for subgraphs of maximum degree 4, 3 and 2. ∗ Was also at Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic. Deceased January 2016. N.H. BONG ET AL. /AUSTRALAS. J. COMBIN. 68 (2) (2017), 245–264 246

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Novi H. Bong

Papers

Some properties of the multiset dimension of graphs

On distance-irregular labelings of cycles and wheels

The threshold strong dimension of a graph

The Threshold Strong Dimension of a Graph.

MaxDDBS problem on butterfly networks.