Realization of distance matrices by unicyclic graphs

Given a distance matrix D, we study the behavior of its compaction vector and reduction matrix with respect to the problem of the realization of D by a weighted graph. To this end, we first give a general result on realization by n−cycles and successively we mainly focus on unicyclic graphs, presenting an algorithm which determines when a distance matrix is realizable by such a kind of graph, and then, shows how to construct it

A New Temporal Interpretation of Cluster Editing

The NP-complete graph problem Cluster Editing seeks to transform a static graph into a disjoint union of cliques by making the fewest possible edits to the edges. We introduce a natural interpretation of this problem in temporal graphs, whose edge sets change over time. This problem is NP-complete even when restricted to temporal graphs whose underlying graph is a path, but we obtain two polynomial-time algorithms for restricted cases. In the static setting, it is well-known that a graph is a disjoint union of cliques if and only if it contains no induced copy of $P_3$; we demonstrate that no general characterisation involving sets of at most four vertices can exist in the temporal setting, but obtain a complete characterisation involving forbidden configurations on at most five vertices. This characterisation gives rise to an FPT algorithm parameterised simultaneously by the permitted number of modifications and the lifetime of the temporal graph.Comment: 26 pages, 2 figures. Extended abstract appeared at IWOCA 202

A New Temporal Interpretation of Cluster Editing

The NP-complete graph problem Cluster Editing seeks to transform a static graph into disjoint union of cliques by making the fewest possible edits to the edge set. We introduce a natural interpreta- tion of this problem in the setting of temporal graphs, whose edge-sets are subject to discrete changes over time, which we call Editing to Tem- poral Cliques. This problem is NP-complete even when restricted to temporal graphs whose underlying graph is a path, but we obtain two polynomial-time algorithms for special cases with further restrictions. In the static setting, it is well-known that a graph is a disjoint union of cliques if and only if it contains no induced copy of P3; we demon- strate that there can be no universal characterisation of cluster temporal graphs in terms of subsets of at most four vertices. However, subject to a minor additional restriction, we obtain a characterisation involving for- bidden configurations on five vertices. This characterisation gives rise to an FPT algorithm parameterised simultaneously by the permitted num- ber of modifications and the lifetime of the temporal graph, which uses a simple search-tree strategy

A New Temporal Interpretation of Cluster Editing

The NP-complete graph problem Cluster Editing seeks to transform a static graph into a disjoint union of cliques by making the fewest possible edits to the edges. We introduce a natural interpretation of this problem in temporal graphs, whose edge sets change over time. This problem is NP-complete even when restricted to temporal graphs whose underlying graph is a path, but we obtain two polynomial-time algorithms for restricted cases. In the static setting, it is well-known that a graph is a disjoint union of cliques if and only if it contains no induced copy of $P_3$; we demonstrate that no general characterisation involving sets of at most four vertices can exist in the temporal setting, but obtain a complete characterisation involving forbidden configurations on at most five vertices. This characterisation gives rise to an FPT algorithm parameterised simultaneously by the permitted number of modifications and the lifetime of the temporal graph