10,989 research outputs found

    Identifying codes in vertex-transitive graphs and strongly regular graphs

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    We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs

    Random subgraphs make identification affordable

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    An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the identifying code number (the size of a smallest identifying code), which indeed is not even a monotone parameter with respect to graph inclusion. We show that every graph GG with nn vertices, maximum degree Δ=ω(1)\Delta=\omega(1) and minimum degree δclogΔ\delta\geq c\log{\Delta}, for some constant c>0c>0, contains a large spanning subgraph which admits an identifying code with size O(nlogΔδ)O\left(\frac{n\log{\Delta}}{\delta}\right). In particular, if δ=Θ(n)\delta=\Theta(n), then GG has a dense spanning subgraph with identifying code O(logn)O\left(\log n\right), namely, of asymptotically optimal size. The subgraph we build is created using a probabilistic approach, and we use an interplay of various random methods to analyze it. Moreover we show that the result is essentially best possible, both in terms of the number of deleted edges and the size of the identifying code

    On the size of identifying codes in triangle-free graphs

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    In an undirected graph GG, a subset CV(G)C\subseteq V(G) such that CC is a dominating set of GG, and each vertex in V(G)V(G) is dominated by a distinct subset of vertices from CC, is called an identifying code of GG. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph GG, let \M(G) be the minimum cardinality of an identifying code in GG. In this paper, we show that for any connected identifiable triangle-free graph GG on nn vertices having maximum degree Δ3\Delta\geq 3, \M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}. This bound is asymptotically tight up to constants due to various classes of graphs including (Δ1)(\Delta-1)-ary trees, which are known to have their minimum identifying code of size nnΔ1+o(1)n-\tfrac{n}{\Delta-1+o(1)}. We also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant cc such that the bound \M(G)\le n-\tfrac{n}{\Delta}+c holds for any nontrivial connected identifiable graph GG

    Bounds and extremal graphs for total dominating identifying codes

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    An identifying code CC of a graph GG is a dominating set of GG such that any two distinct vertices of GG have distinct closed neighbourhoods within CC. The smallest size of an identifying code of GG is denoted γID(G)\gamma^{\text{ID}}(G). When every vertex of GG also has a neighbour in CC, it is said to be a total dominating identifying code of GG, and the smallest size of a total dominating identifying code of GG is denoted by γtID(G)\gamma_t^{\text{ID}}(G). Extending similar characterizations for identifying codes from the literature, we characterize those graphs GG of order nn with γtID(G)=n\gamma_t^{\text{ID}}(G)=n (the only such connected graph is P3P_3) and γtID(G)=n1\gamma_t^{\text{ID}}(G)=n-1 (such graphs either satisfy γID(G)=n1\gamma^{\text{ID}}(G)=n-1 or are built from certain such graphs by adding a set of universal vertices, to each of which a private leaf is attached). Then, using bounds from the literature, we remark that any (open and closed) twin-free tree of order nn has a total dominating identifying code of size at most 3n4\frac{3n}{4}. This bound is tight, and we characterize the trees reaching it. Moreover, by a new proof, we show that this bound actually holds for the larger class of all twin-free graphs of girth at least 5. The cycle C8C_8 also attains this bound. We also provide a generalized bound for all graphs of girth at least 5 (possibly with twins). Finally, we relate γtID(G)\gamma_t^{\text{ID}}(G) to the related parameter γID(G)\gamma^{\text{ID}}(G) as well as the location-domination number of GG and its variants, providing bounds that are either tight or almost tight

    Location-domination in line graphs

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    A set DD of vertices of a graph GG is locating if every two distinct vertices outside DD have distinct neighbors in DD; that is, for distinct vertices uu and vv outside DD, N(u)DN(v)DN(u) \cap D \neq N(v) \cap D, where N(u)N(u) denotes the open neighborhood of uu. If DD is also a dominating set (total dominating set), it is called a locating-dominating set (respectively, locating-total dominating set) of GG. A graph GG is twin-free if every two distinct vertices of GG have distinct open and closed neighborhoods. It is conjectured [D. Garijo, A. Gonzalez and A. Marquez, The difference between the metric dimension and the determining number of a graph. Applied Mathematics and Computation 249 (2014), 487--501] and [F. Foucaud and M. A. Henning. Locating-total dominating sets in twin-free graphs: a conjecture. The Electronic Journal of Combinatorics 23 (2016), P3.9] respectively, that any twin-free graph GG without isolated vertices has a locating-dominating set of size at most one-half its order and a locating-total dominating set of size at most two-thirds its order. In this paper, we prove these two conjectures for the class of line graphs. Both bounds are tight for this class, in the sense that there are infinitely many connected line graphs for which equality holds in the bounds.Comment: 23 pages, 2 figure

    Identifying codes in vertex-transitive graphs and strongly regular graphs

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    We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2 ln(|V|)+1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order |V|^a with a in {1/4,1/3,2/5}. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs

    On three domination numbers in block graphs

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    The problems of determining minimum identifying, locating-dominating or open locating-dominating codes are special search problems that are challenging both from a theoretical and a computational point of view. Hence, a typical line of attack for these problems is to determine lower and upper bounds for minimum codes in special graphs. In this work we study the problem of determining the cardinality of minimum codes in block graphs (that are diamond-free chordal graphs). We present for all three codes lower and upper bounds as well as block graphs where these bounds are attained
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