Weighted Domination Number of Cactus Graphs

Abstract

Abstract:In the paper, we write a linear algorithm for calculating the weighted domination number of a vertexweighted cactus. The algorithm is based on the well known depth first search (DFS) structure. Our algorithm needs less than 12n + 5b additions and 9n + 2b min-operations where n is the number of vertices and b is the number of blocks in the cactus.

The Hosoya polynomial of double weighted graphs

“…Following the idea of [17], we calculate the Hosoya polynomial of some special examples of weighted graphs from the Hosoya polynomials of the given subgraphs. For later reference, auxiliary polynomialsĤ a (G, x), i.e.…”

“…The depth first search (DFS) algorithm is a well known method for exploring graphs. It can be used for recognizing cactus graphs providing the data structure (see [17,21,22,23]). Let G r be a rooted cactus graph with a root r. After running the DFS algorithm, the vertices of G r are DFS ordered.…”

“…We omit detailed description of DFS algorithm here, as it is well known, see for example [21]. The pseudocode of the DFS algorithm is also written in [17]. Some properties of the DFS ordered vertices of cactus and the relationship between the definitions of C, G, H-vertices in a rooted cactus G r and arrays in the DFS table is described in [17].…”

Volume computation for sparse Boolean quadric relaxations

“…Cactus graphs occur in a wide variety of applications, e.g., location theory, communication networks, and stability analysis (see [38] and the references therein). Cactus graphs can be recognized in linear time, via a depth-first search approach (see [38] and [52]), and of course the number of cycles in such a graph is at most n/3. We do note that we can apply Lemma 16, which tells us that odd cycle inequalities from the same odd cycle cut off disjoint parts of Q(G).…”

“…The necklaces N n (see §7) are cactus graphs. Cactus graphs occur in a wide variety of applications, e.g., location theory, communication networks, and stability analysis (see [38] and the references therein). Cactus graphs can be recognized in linear time, via a depth-first search approach (see [38] and [52]), and of course the number of cycles in such a graph is at most n/3.…”