A graph G contains a multigraph H as an induced immersion if H can be obtained from G by a sequence of vertex deletions and lifts. We present a polynomial-time algorithm that decides for any fixed multigraph H whether an input graph G contains H as an induced immersion. We also show that for every multigraph H with maximum degree at most 2, there exists a constant cH such that every graph with ...

Let u; v be distin t verti es of a multigraph G with degrees d u and d v , respe tively. The number of edge-disjoint u; v-paths in G is bounded above by minfd u ; d v g. A multigraph G is optimally edgeonne ted if for all pairs of distin t verti es u and v this upper bound is a hieved. If G is a multigraph with degree sequen e D, then we say G is a realisation of D. We hara terize degree sequen...

Proof of Lemma 1.1. Let H be the sub-multigraph of G consisting only of the vertices and the edges lying on at least one of the paths u and v. Then, the multigraph H is connected, and the paths u and v are two paths in H. Hence, there exist two distinct paths from x to y in H (namely, u and v). Notice that the multigraph H has at most k + ` edges (since its edges consist of the k edges of the p...

Given a family F of multigraphs without isolated vertices, a multigraph M is called F-decomposable if M is an edge disjoint union of multigraphs each of which is isomorphic to a member of F . We present necessary and sufficient conditions for existence of such decompositions if F consists of all multigraphs of size q except for one. Namely, for a multigraph H of size q we find each multigraph M...

Recently, a new set of multigraph parameters was defined, called “gonalities”. Gonality bears some similarity to treewidth, and is a relevant graph parameter for problems in number theory and multigraph algorithms. Multigraphs of gonality 1 are trees. We consider so-called “hyperelliptic graphs” (multigraphs of gonality 2) and provide a safe and complete sets of reduction rules for such multigr...

In this article we build a linear representation starting from multigraph; allows us to give an algebraic view of the multigraph are studying. We show that two isomorphic multigraphs equivalent representations; conversely representations multigraphs. For clarity at beginning, classical results on representations, nevertheless these specific our graph representation.

Let G be a c-chromatic multigraph (c >t 2) with maximum edge multiplicity s. In this note we show that G has an embedding as an induced subgraph, into some degree irregular c-chromatic multigraph having the same maximum edge multiplicity.

How can we accurately estimate local triangles for all nodes in simple and multigraph streams? Local triangle counting in a graph stream is one of the most fundamental tasks in graph mining with important applications including anomaly detection and social network analysis. Although there have been several local triangle counting methods in a graph stream, their estimation has a large variance ...

Let Ka,b be a complete bipartite multigraph. We show a necessary and sufficient condition for a multigraph Ka,b to be arbitrarily decomposable into open trails. A motivation for us to investigated this problem is a paper of Balister [1].

RDF is a standard for the conceptual description of knowledge, and SPARQL is the query language conceived to query RDF data. The RDF data is cherished and exploited by various domains such as life sciences, Semantic Web, social network, etc. Further, its integration at Web-scale compels RDF management engines to deal with complex queries in terms of both size and structure. In this paper, we pr...