Generating minimal redundant and maximal irredundant sets in incidence graphs

Abstract

It has been proved by Boros and Makino that there is no output-polynomial-time algorithm enumerating the minimal redundant sets or the maximal irredundant sets of a hypergraph, unless P=NP. The same question was left open for graphs, with only a few tractable cases known to date. In this paper, we focus on graph classes that capture incidence relations such as bipartite, co-bipartite, and split graphs. Concerning maximal irredundant sets, we show that the problem on co-bipartite graphs is as hard as in general graphs and tractable in split and strongly orderable graphs, the latter being a generalization of chordal bipartite graphs. As for minimal redundant sets enumeration, we first show that the problem is intractable in split and co-bipartite graphs, answering the aforementioned open question, and that it is tractable on $(C_3,C_5,C_6,C_8)$-free graphs, a class of graphs incomparable to strongly orderable graphs, and which also generalizes chordal bipartite graphs.

Figures (8)

• Figure 1: A graph $G$ in which the vertex $v$ is quasi-simple. Within its neighborhood, we have $u_1 \leq u_2 \leq u_3 \leq u_4$ where $\leq$ indicates comparability.
• Figure 2: An hypergraph $\mathcal{H}$ and its incidence graphs $B(\mathcal{H})$, $C(\mathcal{H})$, $S_1(\mathcal{H})$ and $S_2(\mathcal{H})$. In graphs, cliques are indicated by dotted grey zones.
• Figure 3: The reduction of Theorem \ref{['thm:MIRR:cobip']}. On the left a graph $G$ where shaded vertices indicate a maximal irredundant set. On the right the incidence co-bipartite graph $C(\mathcal{N}(G))$ of $\mathcal{N}(G)$. Grey dotted zones indicate the clique bipartition and the shaded vertices form the maximal irredundant set $C(\mathcal{N}(G))$ induced by the maximal irredundant set in pictured in $G$. The vertices boxed in blue illustrate how $N[v_5]$ (in $G$) is encoded in $C(\mathcal{N}(G))$.
• Figure 4: The situations of \ref{['lemma:MRED_cases_two_redundant_vertices']}. In each case, a minimal redundant set is indicated by shaded vertices with (yellow) bold vertices being redundant. The top-left, top-right, and bottom-left cases illustrate that $R$ intersects at most one of $N(x) \setminus \{y\}$ or $N(y) \setminus \{x\}$. The bottom-right graph illustrates one of the two cases of \ref{['lemma:MRED_redundant_contained_closed_neighborhood_of_x']}, where $y$ is also redundant: $R = N[y]$. The other case, not pictured here, is $R = N[x]$.
• Figure 5: Illustration of Lemma \ref{['lemma:MRED:charac_1_redundant_1_neighbor']} on the left, and Lemma \ref{['lemma:MRED:charac_1_redundant_multi_neighbor']} on the right. Shaded vertices indicate minimal redundant sets, and $x$ (in bold yellow) is the redundant vertex in each case. On the left, $y$ (boxed in green) has a private vertex as well as each vertex of $S$ adjacent to $y$ (the vertices of $S$ are boxed in red) and $S$ minimally dominates $N(x) \setminus \{y\}$ (boxed in blue). On the right, $Y$ has at least two elements, $S$ contains no vertex adjacent to $Y$ and minimally dominates $N(x) \setminus Y$.

Theorems & Definitions (63)

• Theorem 1: Reformulation of boros2024generating
• Theorem 3: bodlaender2015open
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Proposition 8
• Lemma 9
• proof
• Proposition 10