Vertex-weighted Graph Research Articles

Finding a Heaviest Vertex-Weighted Triangle Is not Harder than Matrix Multiplication

We show that a maximum-weight triangle in an undirected graph with n vertices and real weights assigned to vertices can be found in time $\mathcal{O}(n^{\omega}+n^{2+o(1)})$, where $\omega$ is the exponent of the fastest matrix multiplication algorithm. By the currently best bound on $\omega$, the running time of our algorithm is $\mathcal{O}(n^{2.376})$. Our algorithm substantially improves the previous time-bounds for this problem, and its asymptotic time complexity matches that of the fastest known algorithm for finding any triangle (not necessarily a maximum-weight one) in a graph. We can extend our algorithm to improve the upper bounds on finding a maximum-weight triangle in a sparse graph and on finding a maximum-weight subgraph isomorphic to a fixed graph. We can find a maximum-weight triangle in a vertex-weighted graph with m edges in asymptotic time required by the fastest algorithm for finding any triangle in a graph with m edges, i.e., in time $\mathcal{O}(m^{1.41})$. Our algorithms for a maximum-weight fixed subgraph (in particular any clique of constant size) are asymptotically as fast as the fastest known algorithms for a fixed subgraph.

On the complexity of the multicut problem in bounded tree-width graphs and digraphs

Given an edge- or vertex-weighted graph or digraph and a list of source–sink pairs, the minimum multicut problem consists in selecting a minimum weight set of edges or vertices whose removal leaves no path from each source to the corresponding sink. This is a classical NP -hard problem, and we show that the edge version becomes tractable in bounded tree-width graphs if the number of source–sink pairs is fixed, but remains NP -hard in directed acyclic graphs and APX -hard in bounded tree-width and bounded degree unweighted digraphs. The vertex version, although tractable in trees, is proved to be NP -hard in unweighted cacti of bounded degree and bounded path-width.

Minimal invariant sets in a vertex-weighted graph

A weighting of vertices of a graph is admissible if there exists an edge weighting such that the weight of each vertex equals the sum of weights of its incident edges. Given an admissible vertex weighting of a graph, an invariant set is an edge set such that the sum of the weights of its edges is the same for every edge weighting, and a nonempty invariant set is minimal if none of its nonempty proper subsets is an invariant set. It is easily seen that every nonempty invariant set is a disjoint union of minimal invariant sets. A graphical characterisation of minimal invariant sets in a bipartite graph is known both in the case the vertex weights are reals and in the case the vertex weights are nonnegative reals. We shall state a graphical characterisation of minimal invariant sets in an arbitrary vertex-weighted graph. Moreover, we give a linear algorithm to test an invariant set for minimality. Finally, we state a complete axiomatisation of invariant sets and give a polynomial algorithm to find a set of minimal invariant sets that completely characterise the set of all invariant sets.

Weighted Coloring: further complexity and approximability results

Given a vertex-weighted graph G = ( V , E ; w ) , w ( v ) ⩾ 0 for any v ∈ V , we consider a weighted version of the coloring problem which consists in finding a partition S = ( S 1 , … , S k ) of the vertex set of G into stable sets and minimizing ∑ i = 1 k w ( S i ) where the weight of S is defined as max { w ( v ) : v ∈ S } . In this paper, we continue the investigation of the complexity and the approximability of this problem by answering some of the questions raised by Guan and Zhu [D.J. Guan, X. Zhu, A coloring problem for weighted graphs, Inform. Process. Lett. 61 (2) (1997) 77–81].

Wiener number of vertex-weighted graphs and a chemical application

The Wiener number W( G) of a graph G is the sum of distances between all pairs of vertices of G. If ( G, w) is a vertex-weighted graph, then the Wiener number W( G, w) of ( G, w) is the sum, over all pairs of vertices, of products of weights of the vertices and their distance. For G being a partial binary Hamming graph, a formula is given for computing W( G, w) in terms of a binary Hamming labeling of G. This result is applied to prove that W(PH) = W( H ̃ S) + 36W(ID) , where PH is a phenylene, H ̃ S a pertinently vertex-weighted hexagonal squeeze of PH, and ID the inner dual of the hexagonal squeeze.

Finding optimal subgraphs by local search

We consider the local search problem of finding a vertex induced subgraph on a vertex-weighted graph that satisfies a fixed graph property Π and maximizes the sum of its vertices' weights. We show that the problem is complete for the class PLS of polynomial-time local search problems if Π is any nontrivial and hereditary graph property, such as planar, acyclic, complete, bipartite and chordal.

Sufficient conditions for realizability of vertex-weighted directed graphs

In this paper we present several sufficient conditions for realizability of terminal capacity matrices of vertex-weighted directed graphs. First necessary and sufficient conditions for a matrix to be realizable by a vertex-weighted directed tree are derived and an algorithm for synthesis is given. Next, as an intermediate step, necessary and sufficient conditions for realizability by a vertex-weighted undirected tree are obtained. These results are then combined to give necessary and sufficient conditions for realizability as a vertex-weighted tree containing both directed and undirected branches. As a corollary, we obtain necessary and sufficient conditions for realizability as a vertex-weighted tree that is a directed path. A simple extension of this result gives realizability conditions for a vertex-weighted directed circuit. Finally we develop necessary and sufficient conditions for realizability as a vertex-weighted graph containing no directed circuits.