A graph is k-supereulerian if it has a spanning even subgraph with at most k components. We show that if G is a connected graph and G is the (collapsible) reduction of G, then G is k-supereulerian if and only if G is k-supereulerian. This extends Catlin’s reduction theorem in [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29–44]. For a graph G, l...

In this paper, we propose several integer programming (IP) formulations to exactly solve the minimum-cost λ -edge-connected k-subgraph problem, or the (k,λ )-subgraph problem, based on its graph properties. Special cases of this problem include the well-known k-minimum spanning tree problem (if λ = 1), λ -edgeconnected spanning subgraph problem (if k = |V |) and k-clique problem (if λ = k−1 and...

The bounded degree maximum spanning subgraph problem arising from wireless mesh networks is studied here. Given a connected graph G and a positive integer d ≥ 2, the problem aims to find a maximum spanning subgraph H of G with the constraint: for every vertex v of G, the degree of v in H, dH(v), is less than or equal to d. Here, a spanning subgraph is a connected subgraph which contains all the...

A graph drawn on the plane is 2-immersed in if each edge crossed by at most two other edges (this drawing of called a 2-immersion plane). 2-planar it can be plane. We consider class T all finite planar graphs triangulating such that have no loops and multiple edges, vertices degree 5 6 only, distance between any 5-valent least 4. describe possible 2-immersions for 2-immersions, we ways which ne...

A directed graph $G=(V,E)$ is strongly biconnected if $G$ connected and its underlying biconnected. called $2$-vertex-strongly $|V|\geq 3$ the induced subgraph on $V\setminus\left\lbrace w\right\rbrace $ for every vertex $w\in V$. In this paper we study following problem. Given a $G=(V,E)$, compute an edge subset $E^{2sb} \subseteq E$ of minimum size such that $(V,E^{2sb})$

By Petersen’s theorem, a bridgeless cubic graph has a 2-factor. H. Fleischner extended this result to bridgeless graphs of minimum degree at least three by showing that every such graph has a spanning even subgraph. Our main result is that, under the stronger hypothesis of 3-edge-connectivity, we can find a spanning even subgraph in which every component has at least five vertices. We show that...

A Halin graph, defined by Halin [3], is a plane graph H = T ∪ C such that T is a spanning tree of H with no vertices of degree 2 where |T | ≥ 4 and C is a cycle whose vertex set is the set of leaves of T . In his work, as an example of a class of edge-minimal 3-connected plane graphs, Halin constructed this family of plane graphs, which have many interesting properties. Lovász and Plummer [5] n...

A graph G is generically [4] rigid in dimension one if and only if it contains a spanning tree, that is, a spanning subgraph assembled by inductively joining 1-simplices along 0-simplices. The analogous property is sufficient but not necessary for the generic rigidity of graphs in higher dimensions, that is, a generically rigid graph in R need not contain a spanning subgraph consisting of n-sim...

This note concerns the (1, f)-odd subgraph problem, i.e. we are given an undirected graph G and an odd value function f : V (G) → N, and our goal is to find a spanning subgraph F of G with degF ≤ f minimizing the number of even degree vertices. First we prove a Gallai–Edmonds type structure theorem and some other known results on the (1, f)-odd subgraph problem, using an easy reduction to the m...

Wepresent e cient parallel algorithms for the problems of nding a minimal2-edge-connected spanning subgraph of a 2-edge-connected graph and nding a minimalbiconnected spanning subgraph of a biconnected graph. The parallel algorithms run in polylog time using a linear number of PRAM processors. We also give linear time sequential algorithms for minimally augmenting a spanning tree into a 2-edge-...