Semi-dynamic algorithms for strongly chordal graphs

Within the broad ambit of algorithm design, the study of dynamic graph algorithms continues to be a thriving area of research. Commensurate with this interest is an extensive literature on the topic. Not surprisingly, dynamic algorithms for all varieties of shortest path problems, in view of their practical importance, occupy a preeminent position. Relevant to this paper are fully dynamic algorithms for chordal graphs. Surprisingly, to the best of our knowledge, there seems to be no reported results for the problem of dynamic algorithms for strongly chordal graphs. To redress this gap, in this paper, we propose a semi-dynamic algorithm for edge-deletions and a semi-dynamic algorithm for edge-insertions in a strongly chordal graph, [Formula: see text]. The query complexity of an edge-deletion is [Formula: see text], where [Formula: see text] and [Formula: see text] are the degrees of the vertices [Formula: see text] and [Formula: see text] of the candidate edge [Formula: see text], while the query complexity of an edge-insertion is [Formula: see text], where [Formula: see text] is the number of vertices of [Formula: see text].

Odd twists on strongly chordal graphs

Strongly chordal graphs can be characterized as chordal graphs in which every even cycle of length at least [Formula: see text] has an odd chord (a chord whose endpoints are an odd distance apart in the cycle subgraph). Define “oddly chordal graphs” to be chordal graphs in which every odd cycle of length at least [Formula: see text] has an odd chord. Strongly chordal graphs are shown to be oddly chordal, and the oddly chordal graphs are characterized by forbidding induced “double [Formula: see text]-sun” subgraphs. Both strongly chordal and oddly chordal graphs are also characterized in terms of uncrossed chords of appropriate-length cycles.

Strengthening strongly chordal graphs

An [Formula: see text]-chord of a cycle [Formula: see text] is a chord that forms a new cycle with a length-[Formula: see text] subpath of [Formula: see text] when [Formula: see text] is at most half the length of [Formula: see text]. Define a graph to be [Formula: see text]-strongly chordal if, for every [Formula: see text], every cycle long enough to have an [Formula: see text]-chord always has an [Formula: see text]-chord. The [Formula: see text]-strongly chordal and [Formula: see text]-strongly chordal graphs are, respectively, the chordal and strongly chordal graphs. Several characterizations of [Formula: see text]-strongly chordal graphs are proved, along with details of the class of [Formula: see text]-strongly chordal graphs.