—In this paper, a Connected Dominating Set (CDS) construction algorithm CSCDS (Connected Subset based CDS) is proposed, which is based on the connected subset concept. The CSCDS contains two main stages, which are dominating set construction stage and connected dominating set construction stage respectively. In the first stage, the dominators are proposed based on the one hop white neighbor in...

A function f :V (G) → {−1, 0, 1} defined on the vertices of a graph G is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. An MTDF f is minimal if there does not exist an MTDF g:V (G) → {−1, 0, 1}, f = g, for which g(v) f (v) for every v ∈ V (G). The weight of an MTDF is the sum of its function values over all vertices. The mi...

1. Solutions. When von Neumann and Morgenstern first defined the solution of a cooperative game, they did so in the context of characteristic function games with side payments. They defined a solution to be a set S of imputations such that: (a) " N o / contained in S is dominated by an x contained in S". (b) "Every/ not contained in S is dominated by some x contained in S". Condition (a) is cal...

We perform a frequentist analysis of the bispectrum of WMAP first year data. We find clear signal domination up to l ≈ 200, with overall consistency with Gaussianity, except for the following features. There is a flat patch (i.e. a low χ region) in the same-l components of the bispectrum spanning the range l = 32 − 62; this may be interpreted as ruling out Gaussianity at the 99.6% confidence le...

A bicriteria network is an interlinked data set where edges are labeled with two cost attributes. An example is a road network where edges represent road segments being labeled with traversal time and energy consumption. To measure the proximity of two nodes in network data, the common method is to compute a cost optimal path between the nodes. In a bicriteria network, there often is no unique ...

A set D ⊆ V (G) of a graph G = (V,E) is a liar’s dominating set if (1) for all v ∈ V (G) |N [v] ∩ D| ≥ 2 and (2) for every pair u, v ∈ V (G) of distinct vertices, |N [u] ∪ N [v] ∩ D| ≥ 3. In this paper, we consider the liar’s domination number of some middle graphs. Every triple dominating set is a liar’s dominating set and every liar’s dominating set must be a double dominating set. So, the li...

In this paper we study cooperative cost games arising from domination problems on graphs. We introduce three games to model the cost allocation problem and we derive a necessary and sufficient condition for the balancedness of all three games. Furthermore we study concavity of these games.

We present a heuristic for finding a small independent dominating set, D, of cubic graphs. We analyse the performance of this heuristic, which is a random greedy algorithm, on random cubic graphs using differential equations and obtain an upper bound on the expected size of D. A corresponding lower bound is derived by means of a direct expectation argument. We prove that D asymptotically almost...

Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard and that each vertex contains at most one guard). More than one guard is allowed to move in response to an attack. The m-eternal domination numbe...

For a graph G, a signed domination function of G is a two-colouring of the vertices of G with colours +1 and –1 such that the closed neighbourhood of every vertex contains more +1’s than –1’s. This concept is closely related to combinatorial discrepancy theory as shown by Füredi and Mubayi [J. Combin. Theory, Ser. B 76 (1999) 223–239]. The signed domination number of G is the minimum of the sum...