On the End-Vertex Problem of Graph Searches

Beisegel, Jesse; Denkert, Carolin; Köhler, Ekkehard; Krnc, Matjaž; Pivač, Nevena; Schleicher, Robert; Strehler, Martin

doi:10.23638/dmtcs-21-1-13

Cited by 4 publications

(7 citation statements)

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“…Corneil et al [11] showed that the end-vertex problem is N P-complete for LBFS on weakly chordal graphs. Similar results were obtained for other searches such as BFS, DFS and MCS [2,7,8], while for several graph classes, among them split graphs, polynomial-time algorithms were presented in [2,7,8,11,15].…”

Section: Introductionsupporting

confidence: 77%

“…From this observation it follows directly that the partial search order problem is N Pcomplete for BFS, DFS, LBFS, LDFS, MCS and MNS [2,8,11].…”

Section: Observation 4 the End-vertex Problem Of A Graph Search A On ...mentioning

confidence: 86%

“…Both the end-vertex problem and the F-tree recognition problem of several searches are well studied on split graphs (see [2,4,8]). In this section we will generalize some of these results to the partial search order problem.…”

Section: Partial Lbfs and Mcs Orders Of Split Graphsmentioning

confidence: 99%

“…This can be done in time O(|π|). To check condition (N 2) in time O(|V (G)|+|E(G)|) we use the algorithm described in the proof of Proposition 14 in [2]. During this process, we can label every vertex in A ∪ N (A) with the index of its I-set or C-set and all other vertices with the label ∞.…”

Section: Partial Lbfs and Mcs Orders Of Split Graphsmentioning

confidence: 99%

“…This result also implies a polynomial-time algorithm for the end-vertex problem on chordal bipartite graphs, a generalization of the result by Gorzny and Huang [15] on the end-vertex problem of LBFS on AT-free bipartite graphs. For split graphs, we will give polynomial-time algorithms for the PSOP of LBFS and MCS that generalize the results on the end-vertex problem [2,8] and the F-tree problem [3] of these searches on this graph class.…”

Section: Introductionmentioning

confidence: 97%

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Linearizing Partial Search Orders

Schleicher¹

2022

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In recent years, questions about the construction of special orderings of a given graph search were studied by several authors. On the one hand, the so called end-vertex problem introduced by Corneil et al. in 2010 asks for search orderings ending in a special vertex. On the other hand, the problem of finding orderings that induce a given search tree was introduced already in the 1980s by Hagerup and received new attention most recently by Beisegel et al. Here, we introduce a generalization of some of these problems by studying the question whether there is a search ordering that is a linear extension of a given partial order on a graph's vertex set. We show that this problem can be solved in polynomial time on chordal bipartite graphs for LBFS, which also implies the first polynomial-time algorithms for the end-vertex problem and two search tree problems for this combination of graph class and search. Furthermore, we present polynomial-time algorithms for LBFS and MCS on split graphs which generalize known results for the end-vertex and search tree problems.

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