In this paper we study three domination-like problems, namely identifying codes, locating-dominating codes, and locating-total-dominating codes. We are interested in finding the minimum cardinality of such codes in circular and infinite grid graphs of given height. We provide an alternate proof for already known results, as well as new results. These were obtained by a computer search based on ...

we study the set of all determinants of adjacency matrices of graphs with a given number of vertices. using brendan mckay's data base of small graphs, determinants of graphs with at most $9$ vertices are computed so that the number of non-isomorphic graphs with given vertices whose determinants are all equal to a number is exhibited in a table. using an idea of m. newman, it is proved that if $...

An open neighbourhood locating-dominating set is a S of vertices graph G such that each vertex has neighbour in S, and for any two u,v G, there at least one exactly u v. We characterize those graphs whose only the whole vertices. More precisely, we prove these are which all connected components half-graphs (a half-graph special bipartite with both parts same size, where part can be ordered so n...

A locating-dominating set D of a graph G = (V (G), E(G)) is a set D ⊆ V (G) such that every vertex of V (G) − D is adjacent to a vertex of D and for every pair of distinct vertices u, v in V (G) − D, N(u) ∩ D = N(v) ∩ D. The minimum cardinality of a locating-dominating set is denoted by γL(G). A graph G is said to be a locating domination edge removal critical graph, or just γ L -ER-critical gr...

The locating-chromatic number of a graph can be defined as the cardinality of a minimum resolving partition of the vertex set such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in are not contained in the same partition class. In this case, the coordinate of a vertex in is expressed in terms of the distances of to all partition classe...

In a graph G = (V, E), an identifying code of G (resp. a locating-dominating code of G) is a subset of vertices C ⊆ V such that N [v]∩C 6= ∅ for all v ∈ V , and N [u] ∩C 6= N [v]∩C for all u 6= v, u, v ∈ V (resp. u, v ∈ V r C), where N [u] denotes the closed neighbourhood of v, that is N [u] = N(u) ∪ {u}. These codes model fault-detection problems in multiprocessor systems and are also used for...

We consider an edge-isoperimetric problem (EIP) on the cartesian powers of graphs. One of our objectives is to extend the list of graphs for whose cartesian powers the lexicographic order provides nested solutions for the EIP. We present several new classes of such graphs that include as special cases all presently known graphs with this property. Our new results are applied to derive best poss...

We introduce nite relational structures called sketches, that represent edge crossings in drawings of nite graphs. We consider the problem of characterizing sketches in Monadic Second-Order logic. We answer positively the question for framed sketches, i.e., for those representing drawings of graphs consisting of a planar connected spanning subgraph (the frame) augmented with additional edges th...

Let c : E(G) → [k] be an edge-coloring of a graph G, not necessarily proper. For each vertex v, let c̄(v) = (a1, . . . , ak), where ai is the number of edges incident to v with color i. Reorder c̄(v) for every v in G in nonincreasing order to obtain c∗(v), the color-blind partition of v. When c∗ induces a proper vertex coloring, that is, c∗(u) 6= c∗(v) for every edge uv in G, we say that c is col...